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Номер 20, страница 46, часть 2 - гдз по алгебре 10 класс учебник Пак, Ардакулы
Авторы: Пак О. В., Ардакулы Д., Ескендирова Е. В.
Тип: Учебник
Издательство: Алматыкітап баспасы
Год издания: 2019 - 2026
Часть: 2
ISBN: 978-601-01-3958-9
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Часть 2. Глава 5. Производная. Параграф 2. Понятие производной. 2.5. Формулы для нахождения производной. Задачи - номер 20, страница 46.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1120179 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "46" "field_page_end" => "47" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/5-2-5zad-20" "field_display_title" => "20" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:5 [ 0 => App\Models\Branch {#1117 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Term {#1118 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1150 #items: array:1 [ 0 => App\Models\Book {#1120 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Term {#1119 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Term {#1124 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1125 #items: array:1 [ 0 => App\Models\Term {#1126 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ …6] "field_translit" => "almatykitap baspasy" ] #original: array:6 [ "id" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ …6] "field_translit" => "almatykitap baspasy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1127 #items: array:3 [ 0 => App\Models\Term {#1128 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "field_surname_rp" => null ] #original: array:12 [ "id" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1129 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "field_surname_rp" => null ] #original: array:12 [ "id" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1130 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1131 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1132 #items: array:1 [ 0 => App\Models\Term {#1133 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1134 #items: array:1 [ 0 => App\Models\Term {#1135 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1136 #items: array:1 [ 0 => App\Models\Term {#1137 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1138 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1139 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1140 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1141 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1142 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1143 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1144 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1146 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1147 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1122} "field_class" => Illuminate\Database\Eloquent\Collection {#1123} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1125} "field_author" => Illuminate\Database\Eloquent\Collection {#1127} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1131} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1132} "field_country" => Illuminate\Database\Eloquent\Collection {#1134} "field_city" => Illuminate\Database\Eloquent\Collection {#1136} "field_series" => Illuminate\Database\Eloquent\Collection {#1138} "field_umk" => Illuminate\Database\Eloquent\Collection {#1139} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1140} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1141} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1142} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1143} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1144} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1145} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1146} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1147} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1148 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1150} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1148} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1149 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119427 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Производная" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1154 #items: array:1 [ 0 => App\Models\Term {#1151 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1153 #items: array:1 [ 0 => App\Models\Book {#1120} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1152 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119427 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Производная" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1154} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1153} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1152} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Branch {#1155 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119431 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Понятие производной" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1159 #items: array:1 [ 0 => App\Models\Term {#1156 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "25" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Book {#1120} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1167 #items: array:1 [ 0 => App\Models\Branch {#1166 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119427 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Производная" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1168 …2} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1169 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1170 …2} ] #original: array:24 [ "id" => 1119427 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Производная" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1168 …2} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1169 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1170 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119431 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Понятие производной" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1159} "field_page_start" => "25" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1158} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1167} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Branch {#1160 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119433 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.5. Формулы для нахождения производной" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1157 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "37" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1163 #items: array:1 [ 0 => App\Models\Book {#1120} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1176 #items: array:1 [ 0 => App\Models\Branch {#1175 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119431 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Понятие производной" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_page_start" => "25" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1178 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1179 …2} ] #original: array:24 [ "id" => 1119431 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Понятие производной" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_page_start" => "25" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1178 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1179 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119433 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.5. Формулы для нахождения производной" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1157} "field_page_start" => "37" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1163} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1176} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Branch {#1164 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119435 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1161 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "44" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1172 #items: array:1 [ 0 => App\Models\Book {#1120} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1185 #items: array:1 [ 0 => App\Models\Branch {#1184 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119433 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.5. Формулы для нахождения производной" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_page_start" => "37" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1187 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1188 …2} ] #original: array:24 [ "id" => 1119433 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.5. Формулы для нахождения производной" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_page_start" => "37" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1187 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1188 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119435 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1161} "field_page_start" => "44" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1172} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1185} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:2 [ 0 => App\Models\Element {#1097 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1300901 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Edition {#1096 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1120179" "type" => "task" ] "text" => "<p><strong>20.</strong> Найдите производные следующих функций:</p><p><strong>а)</strong></p><p>$f(x)=2\sin\frac{x}{2}$, $g(x)=\frac{7}{3}\sin 3x$, $h(x)=-\frac{1}{7}\sin\left(14x+\frac{\pi}{6}\right)$,</p><p>$u(x)=\frac{1}{5\pi}\sin\left(\frac{\pi}{8}-10\pi x\right)$;</p><p><strong>б)</strong></p><p>$f(x)=-22\cos\frac{x}{11}$, $g(x)=\frac{1}{18}\cos 27x$, $h(x)=-\frac{2}{9}\cos\left(\frac{3x}{2}-\frac{\pi}{7}\right)$,</p><p>$u(x)=-\frac{8}{9}\cos\left(\frac{\pi}{8}-\frac{27\pi x}{16}\right)$;</p><p><strong>в)</strong></p><p>$f(x)=4\tan\frac{x}{4}$, $g(x)=\frac{3}{4}\tan\frac{8x}{9}$, $h(x)=\frac{4}{45}\tan\left(\frac{15x}{2}-\frac{\pi}{9}\right)$,</p><p>$u(x)=\frac{1}{2\pi}\tan\left(\frac{\pi}{5}-5\pi x\right)$;</p><p><strong>г)</strong></p><p>$f(x)=-\frac{1}{3}\cot\frac{x}{3}$, $g(x)=\frac{9}{11}\cot\frac{22x}{3}$, $h(x)=\frac{3}{5}\cot\left(\frac{10x}{9}-\frac{\pi}{21}\right)$,</p><p>$u(x)=-\frac{1}{25\pi}\cot\left(\frac{\pi}{11}-75\pi x\right)$.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "20-1.jpg" "alt" => null "width" => "1384" "height" => 690 "path" => "/media/algebra_10/baspasy-u19/0-5-2-5zad/20-1.webp?ts=1753534819" ] 1 => array:5 [ "name" => "20-2.jpg" "alt" => null "width" => "1404" "height" => 646 "path" => "/media/algebra_10/baspasy-u19/0-5-2-5zad/20-2.webp?ts=1753534819" ] ] ] #original: array:7 [ "id" => 1300901 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087} "task" => array:2 [ "refs" => "1120179" "type" => "task" ] "text" => "<p><strong>20.</strong> Найдите производные следующих функций:</p><p><strong>а)</strong></p><p>$f(x)=2\sin\frac{x}{2}$, $g(x)=\frac{7}{3}\sin 3x$, $h(x)=-\frac{1}{7}\sin\left(14x+\frac{\pi}{6}\right)$,</p><p>$u(x)=\frac{1}{5\pi}\sin\left(\frac{\pi}{8}-10\pi x\right)$;</p><p><strong>б)</strong></p><p>$f(x)=-22\cos\frac{x}{11}$, $g(x)=\frac{1}{18}\cos 27x$, $h(x)=-\frac{2}{9}\cos\left(\frac{3x}{2}-\frac{\pi}{7}\right)$,</p><p>$u(x)=-\frac{8}{9}\cos\left(\frac{\pi}{8}-\frac{27\pi x}{16}\right)$;</p><p><strong>в)</strong></p><p>$f(x)=4\tan\frac{x}{4}$, $g(x)=\frac{3}{4}\tan\frac{8x}{9}$, $h(x)=\frac{4}{45}\tan\left(\frac{15x}{2}-\frac{\pi}{9}\right)$,</p><p>$u(x)=\frac{1}{2\pi}\tan\left(\frac{\pi}{5}-5\pi x\right)$;</p><p><strong>г)</strong></p><p>$f(x)=-\frac{1}{3}\cot\frac{x}{3}$, $g(x)=\frac{9}{11}\cot\frac{22x}{3}$, $h(x)=\frac{3}{5}\cot\left(\frac{10x}{9}-\frac{\pi}{21}\right)$,</p><p>$u(x)=-\frac{1}{25\pi}\cot\left(\frac{\pi}{11}-75\pi x\right)$.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "20-1.jpg" "alt" => null "width" => "1384" "height" => 690 "path" => "/media/algebra_10/baspasy-u19/0-5-2-5zad/20-1.webp?ts=1753534819" ] 1 => array:5 [ "name" => "20-2.jpg" "alt" => null "width" => "1404" "height" => 646 "path" => "/media/algebra_10/baspasy-u19/0-5-2-5zad/20-2.webp?ts=1753534819" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1089 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1551937 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Edition {#1090 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1120179" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>$f(x)=2\sin\frac{x}{2}$<br>$f'(x) = (2\sin\frac{x}{2})' = 2 \cdot \cos(\frac{x}{2}) \cdot (\frac{x}{2})' = 2 \cos(\frac{x}{2}) \cdot \frac{1}{2} = \cos(\frac{x}{2})$.<br>Ответ: $f'(x) = \cos(\frac{x}{2})$.</p><p>$g(x) = \frac{7}{3}\sin 3x$<br>$g'(x) = (\frac{7}{3}\sin 3x)' = \frac{7}{3} \cdot \cos(3x) \cdot (3x)' = \frac{7}{3} \cos(3x) \cdot 3 = 7\cos(3x)$.<br>Ответ: $g'(x) = 7\cos(3x)$.</p><p>$h(x)=-\frac{1}{7}\sin(14x+\frac{\pi}{6})$<br>$h'(x) = (-\frac{1}{7}\sin(14x+\frac{\pi}{6}))' = -\frac{1}{7} \cdot \cos(14x+\frac{\pi}{6}) \cdot (14x+\frac{\pi}{6})' = -\frac{1}{7} \cos(14x+\frac{\pi}{6}) \cdot 14 = -2\cos(14x+\frac{\pi}{6})$.<br>Ответ: $h'(x) = -2\cos(14x+\frac{\pi}{6})$.</p><p>$u(x) = \frac{1}{5\pi}\sin(\frac{\pi}{8}-10\pi x)$<br>$u'(x) = (\frac{1}{5\pi}\sin(\frac{\pi}{8}-10\pi x))' = \frac{1}{5\pi} \cdot \cos(\frac{\pi}{8}-10\pi x) \cdot (\frac{\pi}{8}-10\pi x)' = \frac{1}{5\pi} \cos(\frac{\pi}{8}-10\pi x) \cdot (-10\pi) = -2\cos(\frac{\pi}{8}-10\pi x)$.<br>Ответ: $u'(x) = -2\cos(\frac{\pi}{8}-10\pi x)$.</p><p><strong>б)</strong></p><p>$f(x)=-22\cos\frac{x}{11}$<br>$f'(x) = (-22\cos\frac{x}{11})' = -22 \cdot (-\sin\frac{x}{11}) \cdot (\frac{x}{11})' = 22\sin(\frac{x}{11}) \cdot \frac{1}{11} = 2\sin(\frac{x}{11})$.<br>Ответ: $f'(x) = 2\sin(\frac{x}{11})$.</p><p>$g(x) = \frac{1}{18}\cos 27x$<br>$g'(x) = (\frac{1}{18}\cos 27x)' = \frac{1}{18} \cdot (-\sin 27x) \cdot (27x)' = -\frac{1}{18}\sin(27x) \cdot 27 = -\frac{27}{18}\sin(27x) = -\frac{3}{2}\sin(27x)$.<br>Ответ: $g'(x) = -\frac{3}{2}\sin(27x)$.</p><p>$h(x)=-\frac{2}{9}\cos(\frac{3x}{2}-\frac{\pi}{7})$<br>$h'(x) = (-\frac{2}{9}\cos(\frac{3x}{2}-\frac{\pi}{7}))' = -\frac{2}{9} \cdot (-\sin(\frac{3x}{2}-\frac{\pi}{7})) \cdot (\frac{3x}{2}-\frac{\pi}{7})' = \frac{2}{9}\sin(\frac{3x}{2}-\frac{\pi}{7}) \cdot \frac{3}{2} = \frac{1}{3}\sin(\frac{3x}{2}-\frac{\pi}{7})$.<br>Ответ: $h'(x) = \frac{1}{3}\sin(\frac{3x}{2}-\frac{\pi}{7})$.</p><p>$u(x)=-\frac{8}{9}\cos(\frac{\pi}{8}-\frac{27\pi x}{16})$<br>$u'(x) = (-\frac{8}{9}\cos(\frac{\pi}{8}-\frac{27\pi x}{16}))' = -\frac{8}{9} \cdot (-\sin(\frac{\pi}{8}-\frac{27\pi x}{16})) \cdot (\frac{\pi}{8}-\frac{27\pi x}{16})' = \frac{8}{9}\sin(\frac{\pi}{8}-\frac{27\pi x}{16}) \cdot (-\frac{27\pi}{16}) = -\frac{3\pi}{2}\sin(\frac{\pi}{8}-\frac{27\pi x}{16})$.<br>Ответ: $u'(x) = -\frac{3\pi}{2}\sin(\frac{\pi}{8}-\frac{27\pi x}{16})$.</p><p><strong>в)</strong></p><p>$f(x)=4\text{tg}\frac{x}{4}$<br>$f'(x) = (4\text{tg}\frac{x}{4})' = 4 \cdot \frac{1}{\cos^2(\frac{x}{4})} \cdot (\frac{x}{4})' = \frac{4}{\cos^2(\frac{x}{4})} \cdot \frac{1}{4} = \frac{1}{\cos^2(\frac{x}{4})}$.<br>Ответ: $f'(x) = \frac{1}{\cos^2(\frac{x}{4})}$.</p><p>$g(x) = \frac{3}{4}\text{tg}\frac{8x}{9}$<br>$g'(x) = (\frac{3}{4}\text{tg}\frac{8x}{9})' = \frac{3}{4} \cdot \frac{1}{\cos^2(\frac{8x}{9})} \cdot (\frac{8x}{9})' = \frac{3}{4\cos^2(\frac{8x}{9})} \cdot \frac{8}{9} = \frac{2}{3\cos^2(\frac{8x}{9})}$.<br>Ответ: $g'(x) = \frac{2}{3\cos^2(\frac{8x}{9})}$.</p><p>$h(x) = \frac{4}{45}\text{tg}(\frac{15x}{2}-\frac{\pi}{9})$<br>$h'(x) = (\frac{4}{45}\text{tg}(\frac{15x}{2}-\frac{\pi}{9}))' = \frac{4}{45} \cdot \frac{1}{\cos^2(\frac{15x}{2}-\frac{\pi}{9})} \cdot (\frac{15x}{2}-\frac{\pi}{9})' = \frac{4}{45\cos^2(\frac{15x}{2}-\frac{\pi}{9})} \cdot \frac{15}{2} = \frac{2}{3\cos^2(\frac{15x}{2}-\frac{\pi}{9})}$.<br>Ответ: $h'(x) = \frac{2}{3\cos^2(\frac{15x}{2}-\frac{\pi}{9})}$.</p><p>$u(x) = \frac{1}{2\pi}\text{tg}(\frac{\pi}{5}-5\pi x)$<br>$u'(x) = (\frac{1}{2\pi}\text{tg}(\frac{\pi}{5}-5\pi x))' = \frac{1}{2\pi} \cdot \frac{1}{\cos^2(\frac{\pi}{5}-5\pi x)} \cdot (\frac{\pi}{5}-5\pi x)' = \frac{1}{2\pi\cos^2(\frac{\pi}{5}-5\pi x)} \cdot (-5\pi) = -\frac{5}{2\cos^2(\frac{\pi}{5}-5\pi x)}$.<br>Ответ: $u'(x) = -\frac{5}{2\cos^2(\frac{\pi}{5}-5\pi x)}$.</p><p><strong>г)</strong></p><p>$f(x)=-\frac{1}{3}\text{ctg}\frac{x}{3}$<br>$f'(x) = (-\frac{1}{3}\text{ctg}\frac{x}{3})' = -\frac{1}{3} \cdot (-\frac{1}{\sin^2(\frac{x}{3})}) \cdot (\frac{x}{3})' = \frac{1}{3\sin^2(\frac{x}{3})} \cdot \frac{1}{3} = \frac{1}{9\sin^2(\frac{x}{3})}$.<br>Ответ: $f'(x) = \frac{1}{9\sin^2(\frac{x}{3})}$.</p><p>$g(x) = \frac{9}{11}\text{ctg}\frac{22x}{3}$<br>$g'(x) = (\frac{9}{11}\text{ctg}\frac{22x}{3})' = \frac{9}{11} \cdot (-\frac{1}{\sin^2(\frac{22x}{3})}) \cdot (\frac{22x}{3})' = -\frac{9}{11\sin^2(\frac{22x}{3})} \cdot \frac{22}{3} = -\frac{6}{\sin^2(\frac{22x}{3})}$.<br>Ответ: $g'(x) = -\frac{6}{\sin^2(\frac{22x}{3})}$.</p><p>$h(x) = \frac{3}{5}\text{ctg}(\frac{10x}{9}-\frac{\pi}{21})$<br>$h'(x) = (\frac{3}{5}\text{ctg}(\frac{10x}{9}-\frac{\pi}{21}))' = \frac{3}{5} \cdot (-\frac{1}{\sin^2(\frac{10x}{9}-\frac{\pi}{21})}) \cdot (\frac{10x}{9}-\frac{\pi}{21})' = -\frac{3}{5\sin^2(\frac{10x}{9}-\frac{\pi}{21})} \cdot \frac{10}{9} = -\frac{2}{3\sin^2(\frac{10x}{9}-\frac{\pi}{21})}$.<br>Ответ: $h'(x) = -\frac{2}{3\sin^2(\frac{10x}{9}-\frac{\pi}{21})}$.</p><p>$u(x) = -\frac{1}{25\pi}\text{ctg}(\frac{\pi}{11}-75\pi x)$<br>$u'(x) = (-\frac{1}{25\pi}\text{ctg}(\frac{\pi}{11}-75\pi x))' = -\frac{1}{25\pi} \cdot (-\frac{1}{\sin^2(\frac{\pi}{11}-75\pi x)}) \cdot (\frac{\pi}{11}-75\pi x)' = \frac{1}{25\pi\sin^2(\frac{\pi}{11}-75\pi x)} \cdot (-75\pi) = -\frac{3}{\sin^2(\frac{\pi}{11}-75\pi x)}$.<br>Ответ: $u'(x) = -\frac{3}{\sin^2(\frac{\pi}{11}-75\pi x)}$.</p>" ] #original: array:6 [ "id" => 1551937 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081} "task" => array:2 [ "refs" => "1120179" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>$f(x)=2\sin\frac{x}{2}$<br>$f'(x) = (2\sin\frac{x}{2})' = 2 \cdot \cos(\frac{x}{2}) \cdot (\frac{x}{2})' = 2 \cos(\frac{x}{2}) \cdot \frac{1}{2} = \cos(\frac{x}{2})$.<br>Ответ: $f'(x) = \cos(\frac{x}{2})$.</p><p>$g(x) = \frac{7}{3}\sin 3x$<br>$g'(x) = (\frac{7}{3}\sin 3x)' = \frac{7}{3} \cdot \cos(3x) \cdot (3x)' = \frac{7}{3} \cos(3x) \cdot 3 = 7\cos(3x)$.<br>Ответ: $g'(x) = 7\cos(3x)$.</p><p>$h(x)=-\frac{1}{7}\sin(14x+\frac{\pi}{6})$<br>$h'(x) = (-\frac{1}{7}\sin(14x+\frac{\pi}{6}))' = -\frac{1}{7} \cdot \cos(14x+\frac{\pi}{6}) \cdot (14x+\frac{\pi}{6})' = -\frac{1}{7} \cos(14x+\frac{\pi}{6}) \cdot 14 = -2\cos(14x+\frac{\pi}{6})$.<br>Ответ: $h'(x) = -2\cos(14x+\frac{\pi}{6})$.</p><p>$u(x) = \frac{1}{5\pi}\sin(\frac{\pi}{8}-10\pi x)$<br>$u'(x) = (\frac{1}{5\pi}\sin(\frac{\pi}{8}-10\pi x))' = \frac{1}{5\pi} \cdot \cos(\frac{\pi}{8}-10\pi x) \cdot (\frac{\pi}{8}-10\pi x)' = \frac{1}{5\pi} \cos(\frac{\pi}{8}-10\pi x) \cdot (-10\pi) = -2\cos(\frac{\pi}{8}-10\pi x)$.<br>Ответ: $u'(x) = -2\cos(\frac{\pi}{8}-10\pi x)$.</p><p><strong>б)</strong></p><p>$f(x)=-22\cos\frac{x}{11}$<br>$f'(x) = (-22\cos\frac{x}{11})' = -22 \cdot (-\sin\frac{x}{11}) \cdot (\frac{x}{11})' = 22\sin(\frac{x}{11}) \cdot \frac{1}{11} = 2\sin(\frac{x}{11})$.<br>Ответ: $f'(x) = 2\sin(\frac{x}{11})$.</p><p>$g(x) = \frac{1}{18}\cos 27x$<br>$g'(x) = (\frac{1}{18}\cos 27x)' = \frac{1}{18} \cdot (-\sin 27x) \cdot (27x)' = -\frac{1}{18}\sin(27x) \cdot 27 = -\frac{27}{18}\sin(27x) = -\frac{3}{2}\sin(27x)$.<br>Ответ: $g'(x) = -\frac{3}{2}\sin(27x)$.</p><p>$h(x)=-\frac{2}{9}\cos(\frac{3x}{2}-\frac{\pi}{7})$<br>$h'(x) = (-\frac{2}{9}\cos(\frac{3x}{2}-\frac{\pi}{7}))' = -\frac{2}{9} \cdot (-\sin(\frac{3x}{2}-\frac{\pi}{7})) \cdot (\frac{3x}{2}-\frac{\pi}{7})' = \frac{2}{9}\sin(\frac{3x}{2}-\frac{\pi}{7}) \cdot \frac{3}{2} = \frac{1}{3}\sin(\frac{3x}{2}-\frac{\pi}{7})$.<br>Ответ: $h'(x) = \frac{1}{3}\sin(\frac{3x}{2}-\frac{\pi}{7})$.</p><p>$u(x)=-\frac{8}{9}\cos(\frac{\pi}{8}-\frac{27\pi x}{16})$<br>$u'(x) = (-\frac{8}{9}\cos(\frac{\pi}{8}-\frac{27\pi x}{16}))' = -\frac{8}{9} \cdot (-\sin(\frac{\pi}{8}-\frac{27\pi x}{16})) \cdot (\frac{\pi}{8}-\frac{27\pi x}{16})' = \frac{8}{9}\sin(\frac{\pi}{8}-\frac{27\pi x}{16}) \cdot (-\frac{27\pi}{16}) = -\frac{3\pi}{2}\sin(\frac{\pi}{8}-\frac{27\pi x}{16})$.<br>Ответ: $u'(x) = -\frac{3\pi}{2}\sin(\frac{\pi}{8}-\frac{27\pi x}{16})$.</p><p><strong>в)</strong></p><p>$f(x)=4\text{tg}\frac{x}{4}$<br>$f'(x) = (4\text{tg}\frac{x}{4})' = 4 \cdot \frac{1}{\cos^2(\frac{x}{4})} \cdot (\frac{x}{4})' = \frac{4}{\cos^2(\frac{x}{4})} \cdot \frac{1}{4} = \frac{1}{\cos^2(\frac{x}{4})}$.<br>Ответ: $f'(x) = \frac{1}{\cos^2(\frac{x}{4})}$.</p><p>$g(x) = \frac{3}{4}\text{tg}\frac{8x}{9}$<br>$g'(x) = (\frac{3}{4}\text{tg}\frac{8x}{9})' = \frac{3}{4} \cdot \frac{1}{\cos^2(\frac{8x}{9})} \cdot (\frac{8x}{9})' = \frac{3}{4\cos^2(\frac{8x}{9})} \cdot \frac{8}{9} = \frac{2}{3\cos^2(\frac{8x}{9})}$.<br>Ответ: $g'(x) = \frac{2}{3\cos^2(\frac{8x}{9})}$.</p><p>$h(x) = \frac{4}{45}\text{tg}(\frac{15x}{2}-\frac{\pi}{9})$<br>$h'(x) = (\frac{4}{45}\text{tg}(\frac{15x}{2}-\frac{\pi}{9}))' = \frac{4}{45} \cdot \frac{1}{\cos^2(\frac{15x}{2}-\frac{\pi}{9})} \cdot (\frac{15x}{2}-\frac{\pi}{9})' = \frac{4}{45\cos^2(\frac{15x}{2}-\frac{\pi}{9})} \cdot \frac{15}{2} = \frac{2}{3\cos^2(\frac{15x}{2}-\frac{\pi}{9})}$.<br>Ответ: $h'(x) = \frac{2}{3\cos^2(\frac{15x}{2}-\frac{\pi}{9})}$.</p><p>$u(x) = \frac{1}{2\pi}\text{tg}(\frac{\pi}{5}-5\pi x)$<br>$u'(x) = (\frac{1}{2\pi}\text{tg}(\frac{\pi}{5}-5\pi x))' = \frac{1}{2\pi} \cdot \frac{1}{\cos^2(\frac{\pi}{5}-5\pi x)} \cdot (\frac{\pi}{5}-5\pi x)' = \frac{1}{2\pi\cos^2(\frac{\pi}{5}-5\pi x)} \cdot (-5\pi) = -\frac{5}{2\cos^2(\frac{\pi}{5}-5\pi x)}$.<br>Ответ: $u'(x) = -\frac{5}{2\cos^2(\frac{\pi}{5}-5\pi x)}$.</p><p><strong>г)</strong></p><p>$f(x)=-\frac{1}{3}\text{ctg}\frac{x}{3}$<br>$f'(x) = (-\frac{1}{3}\text{ctg}\frac{x}{3})' = -\frac{1}{3} \cdot (-\frac{1}{\sin^2(\frac{x}{3})}) \cdot (\frac{x}{3})' = \frac{1}{3\sin^2(\frac{x}{3})} \cdot \frac{1}{3} = \frac{1}{9\sin^2(\frac{x}{3})}$.<br>Ответ: $f'(x) = \frac{1}{9\sin^2(\frac{x}{3})}$.</p><p>$g(x) = \frac{9}{11}\text{ctg}\frac{22x}{3}$<br>$g'(x) = (\frac{9}{11}\text{ctg}\frac{22x}{3})' = \frac{9}{11} \cdot (-\frac{1}{\sin^2(\frac{22x}{3})}) \cdot (\frac{22x}{3})' = -\frac{9}{11\sin^2(\frac{22x}{3})} \cdot \frac{22}{3} = -\frac{6}{\sin^2(\frac{22x}{3})}$.<br>Ответ: $g'(x) = -\frac{6}{\sin^2(\frac{22x}{3})}$.</p><p>$h(x) = \frac{3}{5}\text{ctg}(\frac{10x}{9}-\frac{\pi}{21})$<br>$h'(x) = (\frac{3}{5}\text{ctg}(\frac{10x}{9}-\frac{\pi}{21}))' = \frac{3}{5} \cdot (-\frac{1}{\sin^2(\frac{10x}{9}-\frac{\pi}{21})}) \cdot (\frac{10x}{9}-\frac{\pi}{21})' = -\frac{3}{5\sin^2(\frac{10x}{9}-\frac{\pi}{21})} \cdot \frac{10}{9} = -\frac{2}{3\sin^2(\frac{10x}{9}-\frac{\pi}{21})}$.<br>Ответ: $h'(x) = -\frac{2}{3\sin^2(\frac{10x}{9}-\frac{\pi}{21})}$.</p><p>$u(x) = -\frac{1}{25\pi}\text{ctg}(\frac{\pi}{11}-75\pi x)$<br>$u'(x) = (-\frac{1}{25\pi}\text{ctg}(\frac{\pi}{11}-75\pi x))' = -\frac{1}{25\pi} \cdot (-\frac{1}{\sin^2(\frac{\pi}{11}-75\pi x)}) \cdot (\frac{\pi}{11}-75\pi x)' = \frac{1}{25\pi\sin^2(\frac{\pi}{11}-75\pi x)} \cdot (-75\pi) = -\frac{3}{\sin^2(\frac{\pi}{11}-75\pi x)}$.<br>Ответ: $u'(x) = -\frac{3}{\sin^2(\frac{\pi}{11}-75\pi x)}$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1120180" "type" => "task" ] "previous" => array:2 [ "refs" => "1120178" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Book {#1120} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1075 #items: array:1 [ 0 => App\Models\BookPage {#1101 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1099199 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "46" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/page-2-46" "field_display_title" => "46" "field_folder" => "2" "field_image_name" => "46" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1083 #items: array:1 [ 0 => App\Models\Branch {#1117} ] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:1 [ 0 => App\Models\Book {#1120} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\Element {#1112 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1262624 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1113 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1262624 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1113 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1099200" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1099198" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1294 #items: array:6 [ 0 => App\Models\Task {#1306 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1120174 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "46" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/5-2-5zad-15" "field_display_title" => "15" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1307 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1308 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1309 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1310 …2} "content" => Illuminate\Database\Eloquent\Collection {#1312 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1318 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1120174 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "46" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/5-2-5zad-15" "field_display_title" => "15" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1307 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null 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№20 (с. 46)
Условие. №20 (с. 46)
скриншот условия
20. Найдите производные следующих функций:
а)
$f(x)=2\sin\frac{x}{2}$, $g(x)=\frac{7}{3}\sin 3x$, $h(x)=-\frac{1}{7}\sin\left(14x+\frac{\pi}{6}\right)$,
$u(x)=\frac{1}{5\pi}\sin\left(\frac{\pi}{8}-10\pi x\right)$;
б)
$f(x)=-22\cos\frac{x}{11}$, $g(x)=\frac{1}{18}\cos 27x$, $h(x)=-\frac{2}{9}\cos\left(\frac{3x}{2}-\frac{\pi}{7}\right)$,
$u(x)=-\frac{8}{9}\cos\left(\frac{\pi}{8}-\frac{27\pi x}{16}\right)$;
в)
$f(x)=4\tan\frac{x}{4}$, $g(x)=\frac{3}{4}\tan\frac{8x}{9}$, $h(x)=\frac{4}{45}\tan\left(\frac{15x}{2}-\frac{\pi}{9}\right)$,
$u(x)=\frac{1}{2\pi}\tan\left(\frac{\pi}{5}-5\pi x\right)$;
г)
$f(x)=-\frac{1}{3}\cot\frac{x}{3}$, $g(x)=\frac{9}{11}\cot\frac{22x}{3}$, $h(x)=\frac{3}{5}\cot\left(\frac{10x}{9}-\frac{\pi}{21}\right)$,
$u(x)=-\frac{1}{25\pi}\cot\left(\frac{\pi}{11}-75\pi x\right)$.
Решение 2 (rus). №20 (с. 46)
а)
$f(x)=2\sin\frac{x}{2}$
$f'(x) = (2\sin\frac{x}{2})' = 2 \cdot \cos(\frac{x}{2}) \cdot (\frac{x}{2})' = 2 \cos(\frac{x}{2}) \cdot \frac{1}{2} = \cos(\frac{x}{2})$.
Ответ: $f'(x) = \cos(\frac{x}{2})$.
$g(x) = \frac{7}{3}\sin 3x$
$g'(x) = (\frac{7}{3}\sin 3x)' = \frac{7}{3} \cdot \cos(3x) \cdot (3x)' = \frac{7}{3} \cos(3x) \cdot 3 = 7\cos(3x)$.
Ответ: $g'(x) = 7\cos(3x)$.
$h(x)=-\frac{1}{7}\sin(14x+\frac{\pi}{6})$
$h'(x) = (-\frac{1}{7}\sin(14x+\frac{\pi}{6}))' = -\frac{1}{7} \cdot \cos(14x+\frac{\pi}{6}) \cdot (14x+\frac{\pi}{6})' = -\frac{1}{7} \cos(14x+\frac{\pi}{6}) \cdot 14 = -2\cos(14x+\frac{\pi}{6})$.
Ответ: $h'(x) = -2\cos(14x+\frac{\pi}{6})$.
$u(x) = \frac{1}{5\pi}\sin(\frac{\pi}{8}-10\pi x)$
$u'(x) = (\frac{1}{5\pi}\sin(\frac{\pi}{8}-10\pi x))' = \frac{1}{5\pi} \cdot \cos(\frac{\pi}{8}-10\pi x) \cdot (\frac{\pi}{8}-10\pi x)' = \frac{1}{5\pi} \cos(\frac{\pi}{8}-10\pi x) \cdot (-10\pi) = -2\cos(\frac{\pi}{8}-10\pi x)$.
Ответ: $u'(x) = -2\cos(\frac{\pi}{8}-10\pi x)$.
б)
$f(x)=-22\cos\frac{x}{11}$
$f'(x) = (-22\cos\frac{x}{11})' = -22 \cdot (-\sin\frac{x}{11}) \cdot (\frac{x}{11})' = 22\sin(\frac{x}{11}) \cdot \frac{1}{11} = 2\sin(\frac{x}{11})$.
Ответ: $f'(x) = 2\sin(\frac{x}{11})$.
$g(x) = \frac{1}{18}\cos 27x$
$g'(x) = (\frac{1}{18}\cos 27x)' = \frac{1}{18} \cdot (-\sin 27x) \cdot (27x)' = -\frac{1}{18}\sin(27x) \cdot 27 = -\frac{27}{18}\sin(27x) = -\frac{3}{2}\sin(27x)$.
Ответ: $g'(x) = -\frac{3}{2}\sin(27x)$.
$h(x)=-\frac{2}{9}\cos(\frac{3x}{2}-\frac{\pi}{7})$
$h'(x) = (-\frac{2}{9}\cos(\frac{3x}{2}-\frac{\pi}{7}))' = -\frac{2}{9} \cdot (-\sin(\frac{3x}{2}-\frac{\pi}{7})) \cdot (\frac{3x}{2}-\frac{\pi}{7})' = \frac{2}{9}\sin(\frac{3x}{2}-\frac{\pi}{7}) \cdot \frac{3}{2} = \frac{1}{3}\sin(\frac{3x}{2}-\frac{\pi}{7})$.
Ответ: $h'(x) = \frac{1}{3}\sin(\frac{3x}{2}-\frac{\pi}{7})$.
$u(x)=-\frac{8}{9}\cos(\frac{\pi}{8}-\frac{27\pi x}{16})$
$u'(x) = (-\frac{8}{9}\cos(\frac{\pi}{8}-\frac{27\pi x}{16}))' = -\frac{8}{9} \cdot (-\sin(\frac{\pi}{8}-\frac{27\pi x}{16})) \cdot (\frac{\pi}{8}-\frac{27\pi x}{16})' = \frac{8}{9}\sin(\frac{\pi}{8}-\frac{27\pi x}{16}) \cdot (-\frac{27\pi}{16}) = -\frac{3\pi}{2}\sin(\frac{\pi}{8}-\frac{27\pi x}{16})$.
Ответ: $u'(x) = -\frac{3\pi}{2}\sin(\frac{\pi}{8}-\frac{27\pi x}{16})$.
в)
$f(x)=4\text{tg}\frac{x}{4}$
$f'(x) = (4\text{tg}\frac{x}{4})' = 4 \cdot \frac{1}{\cos^2(\frac{x}{4})} \cdot (\frac{x}{4})' = \frac{4}{\cos^2(\frac{x}{4})} \cdot \frac{1}{4} = \frac{1}{\cos^2(\frac{x}{4})}$.
Ответ: $f'(x) = \frac{1}{\cos^2(\frac{x}{4})}$.
$g(x) = \frac{3}{4}\text{tg}\frac{8x}{9}$
$g'(x) = (\frac{3}{4}\text{tg}\frac{8x}{9})' = \frac{3}{4} \cdot \frac{1}{\cos^2(\frac{8x}{9})} \cdot (\frac{8x}{9})' = \frac{3}{4\cos^2(\frac{8x}{9})} \cdot \frac{8}{9} = \frac{2}{3\cos^2(\frac{8x}{9})}$.
Ответ: $g'(x) = \frac{2}{3\cos^2(\frac{8x}{9})}$.
$h(x) = \frac{4}{45}\text{tg}(\frac{15x}{2}-\frac{\pi}{9})$
$h'(x) = (\frac{4}{45}\text{tg}(\frac{15x}{2}-\frac{\pi}{9}))' = \frac{4}{45} \cdot \frac{1}{\cos^2(\frac{15x}{2}-\frac{\pi}{9})} \cdot (\frac{15x}{2}-\frac{\pi}{9})' = \frac{4}{45\cos^2(\frac{15x}{2}-\frac{\pi}{9})} \cdot \frac{15}{2} = \frac{2}{3\cos^2(\frac{15x}{2}-\frac{\pi}{9})}$.
Ответ: $h'(x) = \frac{2}{3\cos^2(\frac{15x}{2}-\frac{\pi}{9})}$.
$u(x) = \frac{1}{2\pi}\text{tg}(\frac{\pi}{5}-5\pi x)$
$u'(x) = (\frac{1}{2\pi}\text{tg}(\frac{\pi}{5}-5\pi x))' = \frac{1}{2\pi} \cdot \frac{1}{\cos^2(\frac{\pi}{5}-5\pi x)} \cdot (\frac{\pi}{5}-5\pi x)' = \frac{1}{2\pi\cos^2(\frac{\pi}{5}-5\pi x)} \cdot (-5\pi) = -\frac{5}{2\cos^2(\frac{\pi}{5}-5\pi x)}$.
Ответ: $u'(x) = -\frac{5}{2\cos^2(\frac{\pi}{5}-5\pi x)}$.
г)
$f(x)=-\frac{1}{3}\text{ctg}\frac{x}{3}$
$f'(x) = (-\frac{1}{3}\text{ctg}\frac{x}{3})' = -\frac{1}{3} \cdot (-\frac{1}{\sin^2(\frac{x}{3})}) \cdot (\frac{x}{3})' = \frac{1}{3\sin^2(\frac{x}{3})} \cdot \frac{1}{3} = \frac{1}{9\sin^2(\frac{x}{3})}$.
Ответ: $f'(x) = \frac{1}{9\sin^2(\frac{x}{3})}$.
$g(x) = \frac{9}{11}\text{ctg}\frac{22x}{3}$
$g'(x) = (\frac{9}{11}\text{ctg}\frac{22x}{3})' = \frac{9}{11} \cdot (-\frac{1}{\sin^2(\frac{22x}{3})}) \cdot (\frac{22x}{3})' = -\frac{9}{11\sin^2(\frac{22x}{3})} \cdot \frac{22}{3} = -\frac{6}{\sin^2(\frac{22x}{3})}$.
Ответ: $g'(x) = -\frac{6}{\sin^2(\frac{22x}{3})}$.
$h(x) = \frac{3}{5}\text{ctg}(\frac{10x}{9}-\frac{\pi}{21})$
$h'(x) = (\frac{3}{5}\text{ctg}(\frac{10x}{9}-\frac{\pi}{21}))' = \frac{3}{5} \cdot (-\frac{1}{\sin^2(\frac{10x}{9}-\frac{\pi}{21})}) \cdot (\frac{10x}{9}-\frac{\pi}{21})' = -\frac{3}{5\sin^2(\frac{10x}{9}-\frac{\pi}{21})} \cdot \frac{10}{9} = -\frac{2}{3\sin^2(\frac{10x}{9}-\frac{\pi}{21})}$.
Ответ: $h'(x) = -\frac{2}{3\sin^2(\frac{10x}{9}-\frac{\pi}{21})}$.
$u(x) = -\frac{1}{25\pi}\text{ctg}(\frac{\pi}{11}-75\pi x)$
$u'(x) = (-\frac{1}{25\pi}\text{ctg}(\frac{\pi}{11}-75\pi x))' = -\frac{1}{25\pi} \cdot (-\frac{1}{\sin^2(\frac{\pi}{11}-75\pi x)}) \cdot (\frac{\pi}{11}-75\pi x)' = \frac{1}{25\pi\sin^2(\frac{\pi}{11}-75\pi x)} \cdot (-75\pi) = -\frac{3}{\sin^2(\frac{\pi}{11}-75\pi x)}$.
Ответ: $u'(x) = -\frac{3}{\sin^2(\frac{\pi}{11}-75\pi x)}$.
Другие задания:
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