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Номер 3, страница 44, 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 3, страница 44.
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" => 1120162 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "44" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/5-2-5zad-3" "field_display_title" => "3" "field_outside_task" => null "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 {#1480 #items: array:5 [ 0 => App\Models\Branch {#1070 #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 {#1073 #items: array:1 [ 0 => App\Models\Term {#1071 #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 {#1074 #items: array:1 [ 0 => App\Models\Book {#1075 #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 {#1076 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1098 …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 {#1099 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1102 …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 {#1076 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1098 …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 {#1099 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1102 …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 {#1103 #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 {#1073} "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 {#1074} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1103} ] #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 {#1104 #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 {#1105 #items: array:1 [ 0 => App\Models\Term {#1106 #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 {#1107 #items: array:1 [ 0 => App\Models\Book {#1108 #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 {#1109 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1131 …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 {#1132 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1135 …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 {#1109 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1131 …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 {#1132 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1135 …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 {#1136 #items: array:1 [ 0 => App\Models\Branch {#1137 #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 {#1138 …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 {#1140 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1169 …2} ] #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 {#1138 …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 {#1140 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1169 …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" => 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 {#1105} "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 {#1107} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1136} ] #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 {#1170 #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 {#1171 #items: array:1 [ 0 => App\Models\Term {#1172 #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 {#1173 #items: array:1 [ 0 => App\Models\Book {#1174 #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 {#1175 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1192 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1195 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1196 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1197 …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 {#1198 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1201 …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 {#1175 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1192 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1195 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1196 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1197 …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] …14 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:1 [ 0 => App\Models\Branch {#1203 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #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 {#1171} "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 {#1173} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1202} ] #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 {#1269 #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 {#1270 #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 {#1271 #items: array:1 [ 0 => App\Models\Book {#1272 #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 [ …50] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1300 #items: array:1 [ 0 => App\Models\Branch {#1301 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #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 {#1270} "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 {#1271} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1300} ] #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 {#1372 #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 {#1373 #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 {#1374 #items: array:1 [ 0 => App\Models\Book {#1375 #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 [ …50] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1403 #items: array:1 [ 0 => App\Models\Branch {#1404 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #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 {#1373} "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 {#1374} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1403} ] #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 {#1479 #items: array:2 [ 0 => App\Models\Element {#1490 #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" => 1300884 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1499 #items: array:1 [ 0 => App\Models\Edition {#1491 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1120162" "type" => "task" ] "text" => "<p><strong>3.</strong> Найдите производные следующих функций:</p><p><strong>а)</strong> $f(x)=\sin x$, $g(x)=\sin 2x$, $h(x)=\frac{1}{6}\sin\left(3x-\frac{\pi}{3}\right)$, $u(x)=-\frac{1}{\pi}\sin\left(\frac{\pi}{5}-4\pi x\right)$;</p><p><strong>б)</strong> $f(x)=-\cos x$, $g(x)=\cos 3x$, $h(x)=\frac{1}{4}\cos\left(8x-\frac{\pi}{7}\right)$, $u(x)=-\frac{2}{\pi}\cos\left(\frac{\pi}{8}-\frac{3\pi x}{2}\right)$;</p><p><strong>в)</strong> $f(x)=2\operatorname{tg}x$, $g(x)=\operatorname{tg}\frac{5x}{2}$, $h(x)=\frac{3}{7}\operatorname{tg}\left(\frac{14x}{3}-\frac{\pi}{9}\right)$, $u(x)=-\frac{1}{3}\operatorname{tg}\left(\frac{\pi}{5}-9x\right)$;</p><p><strong>г)</strong> $f(x)=-3\operatorname{ctg}x$, $g(x)=7\operatorname{ctg}\frac{8x}{7}$, $h(x)=-\frac{4}{9}\operatorname{ctg}\left(\frac{27x}{2}-\frac{\pi}{21}\right)$, $u(x)=-\frac{1}{11\pi}\operatorname{ctg}\left(\frac{\pi}{11}-22\pi x\right)$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "3-1.jpg" "alt" => null "width" => "1689" "height" => 845 "path" => "/media/algebra_10/baspasy-u19/0-5-2-5zad/3-1.webp?ts=1753534819" ] ] ] #original: array:7 [ "id" => 1300884 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1499} "task" => array:2 [ "refs" => "1120162" "type" => "task" ] "text" => "<p><strong>3.</strong> Найдите производные следующих функций:</p><p><strong>а)</strong> $f(x)=\sin x$, $g(x)=\sin 2x$, $h(x)=\frac{1}{6}\sin\left(3x-\frac{\pi}{3}\right)$, $u(x)=-\frac{1}{\pi}\sin\left(\frac{\pi}{5}-4\pi x\right)$;</p><p><strong>б)</strong> $f(x)=-\cos x$, $g(x)=\cos 3x$, $h(x)=\frac{1}{4}\cos\left(8x-\frac{\pi}{7}\right)$, $u(x)=-\frac{2}{\pi}\cos\left(\frac{\pi}{8}-\frac{3\pi x}{2}\right)$;</p><p><strong>в)</strong> $f(x)=2\operatorname{tg}x$, $g(x)=\operatorname{tg}\frac{5x}{2}$, $h(x)=\frac{3}{7}\operatorname{tg}\left(\frac{14x}{3}-\frac{\pi}{9}\right)$, $u(x)=-\frac{1}{3}\operatorname{tg}\left(\frac{\pi}{5}-9x\right)$;</p><p><strong>г)</strong> $f(x)=-3\operatorname{ctg}x$, $g(x)=7\operatorname{ctg}\frac{8x}{7}$, $h(x)=-\frac{4}{9}\operatorname{ctg}\left(\frac{27x}{2}-\frac{\pi}{21}\right)$, $u(x)=-\frac{1}{11\pi}\operatorname{ctg}\left(\frac{\pi}{11}-22\pi x\right)$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "3-1.jpg" "alt" => null "width" => "1689" "height" => 845 "path" => "/media/algebra_10/baspasy-u19/0-5-2-5zad/3-1.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 {#1497 #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" => 1551920 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1507 #items: array:1 [ 0 => App\Models\Edition {#1498 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1120162" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Для функции $f(x) = \sin x$ производная является табличной.<br>$f'(x) = (\sin x)' = \cos x$.<br><b>Ответ:</b> $f'(x) = \cos x$.</p><p>Для функции $g(x) = \sin 2x$ применяем правило дифференцирования сложной функции: $(\sin(kx))' = k\cos(kx)$.<br>$g'(x) = (\sin 2x)' = \cos(2x) \cdot (2x)' = 2\cos 2x$.<br><b>Ответ:</b> $g'(x) = 2\cos 2x$.</p><p>Для функции $h(x) = \frac{1}{6}\sin(3x - \frac{\pi}{3})$ применяем правило дифференцирования сложной функции и вынесение константы за знак производной.<br>$h'(x) = \frac{1}{6}(\sin(3x - \frac{\pi}{3}))' = \frac{1}{6} \cos(3x - \frac{\pi}{3}) \cdot (3x - \frac{\pi}{3})' = \frac{1}{6} \cos(3x - \frac{\pi}{3}) \cdot 3 = \frac{3}{6}\cos(3x - \frac{\pi}{3}) = \frac{1}{2}\cos(3x - \frac{\pi}{3})$.<br><b>Ответ:</b> $h'(x) = \frac{1}{2}\cos(3x - \frac{\pi}{3})$.</p><p>Для функции $u(x) = -\frac{1}{\pi}\sin(\frac{\pi}{5} - 4\pi x)$ применяем те же правила.<br>$u'(x) = -\frac{1}{\pi}(\sin(\frac{\pi}{5} - 4\pi x))' = -\frac{1}{\pi}\cos(\frac{\pi}{5} - 4\pi x) \cdot (\frac{\pi}{5} - 4\pi x)' = -\frac{1}{\pi}\cos(\frac{\pi}{5} - 4\pi x) \cdot (-4\pi) = 4\cos(\frac{\pi}{5} - 4\pi x)$.<br><b>Ответ:</b> $u'(x) = 4\cos(\frac{\pi}{5} - 4\pi x)$.</p><p><strong>б)</strong></p><p>Для функции $f(x) = -\cos x$ производная равна:<br>$f'(x) = -(\cos x)' = -(-\sin x) = \sin x$.<br><b>Ответ:</b> $f'(x) = \sin x$.</p><p>Для функции $g(x) = \cos 3x$ применяем правило дифференцирования сложной функции: $(\cos(kx))' = -k\sin(kx)$.<br>$g'(x) = (\cos 3x)' = -\sin(3x) \cdot (3x)' = -3\sin 3x$.<br><b>Ответ:</b> $g'(x) = -3\sin 3x$.</p><p>Для функции $h(x) = \frac{1}{4}\cos(8x - \frac{\pi}{7})$:<br>$h'(x) = \frac{1}{4}(\cos(8x - \frac{\pi}{7}))' = \frac{1}{4}(-\sin(8x - \frac{\pi}{7})) \cdot (8x - \frac{\pi}{7})' = -\frac{1}{4}\sin(8x - \frac{\pi}{7}) \cdot 8 = -2\sin(8x - \frac{\pi}{7})$.<br><b>Ответ:</b> $h'(x) = -2\sin(8x - \frac{\pi}{7})$.</p><p>Для функции $u(x) = \frac{2}{\pi}\cos(\frac{\pi}{8} - \frac{3\pi x}{2})$:<br>$u'(x) = \frac{2}{\pi}(\cos(\frac{\pi}{8} - \frac{3\pi x}{2}))' = \frac{2}{\pi}(-\sin(\frac{\pi}{8} - \frac{3\pi x}{2})) \cdot (\frac{\pi}{8} - \frac{3\pi x}{2})' = -\frac{2}{\pi}\sin(\frac{\pi}{8} - \frac{3\pi x}{2}) \cdot (-\frac{3\pi}{2}) = 3\sin(\frac{\pi}{8} - \frac{3\pi x}{2})$.<br><b>Ответ:</b> $u'(x) = 3\sin(\frac{\pi}{8} - \frac{3\pi x}{2})$.</p><p><strong>в)</strong></p><p>Для функции $f(x) = 2\tan x$, используя табличную производную $(\tan x)' = \frac{1}{\cos^2 x}$:<br>$f'(x) = 2(\tan x)' = \frac{2}{\cos^2 x}$.<br><b>Ответ:</b> $f'(x) = \frac{2}{\cos^2 x}$.</p><p>Для функции $g(x) = \tan \frac{5x}{2}$, используя правило дифференцирования сложной функции:<br>$g'(x) = (\tan \frac{5x}{2})' = \frac{1}{\cos^2(\frac{5x}{2})} \cdot (\frac{5x}{2})' = \frac{1}{\cos^2(\frac{5x}{2})} \cdot \frac{5}{2} = \frac{5}{2\cos^2(\frac{5x}{2})}$.<br><b>Ответ:</b> $g'(x) = \frac{5}{2\cos^2(\frac{5x}{2})}$.</p><p>Для функции $h(x) = \frac{3}{7}\tan(\frac{14x}{3} - \frac{\pi}{9})$:<br>$h'(x) = \frac{3}{7}(\tan(\frac{14x}{3} - \frac{\pi}{9}))' = \frac{3}{7}\frac{1}{\cos^2(\frac{14x}{3} - \frac{\pi}{9})} \cdot (\frac{14x}{3} - \frac{\pi}{9})' = \frac{3}{7}\frac{1}{\cos^2(\frac{14x}{3} - \frac{\pi}{9})} \cdot \frac{14}{3} = \frac{2}{\cos^2(\frac{14x}{3} - \frac{\pi}{9})}$.<br><b>Ответ:</b> $h'(x) = \frac{2}{\cos^2(\frac{14x}{3} - \frac{\pi}{9})}$.</p><p>Для функции $u(x) = -\frac{1}{3}\tan(\frac{\pi}{5} - 9x)$:<br>$u'(x) = -\frac{1}{3}(\tan(\frac{\pi}{5} - 9x))' = -\frac{1}{3}\frac{1}{\cos^2(\frac{\pi}{5} - 9x)} \cdot (\frac{\pi}{5} - 9x)' = -\frac{1}{3}\frac{1}{\cos^2(\frac{\pi}{5} - 9x)} \cdot (-9) = \frac{3}{\cos^2(\frac{\pi}{5} - 9x)}$.<br><b>Ответ:</b> $u'(x) = \frac{3}{\cos^2(\frac{\pi}{5} - 9x)}$.</p><p><strong>г)</strong></p><p>Для функции $f(x) = -3\cot x$, используя табличную производную $(\cot x)' = -\frac{1}{\sin^2 x}$:<br>$f'(x) = -3(\cot x)' = -3(-\frac{1}{\sin^2 x}) = \frac{3}{\sin^2 x}$.<br><b>Ответ:</b> $f'(x) = \frac{3}{\sin^2 x}$.</p><p>Для функции $g(x) = 7\cot \frac{8x}{7}$, используя правило дифференцирования сложной функции:<br>$g'(x) = 7(\cot \frac{8x}{7})' = 7(-\frac{1}{\sin^2(\frac{8x}{7})}) \cdot (\frac{8x}{7})' = -7\frac{1}{\sin^2(\frac{8x}{7})} \cdot \frac{8}{7} = -\frac{8}{\sin^2(\frac{8x}{7})}$.<br><b>Ответ:</b> $g'(x) = -\frac{8}{\sin^2(\frac{8x}{7})}$.</p><p>Для функции $h(x) = \frac{4}{9}\cot(\frac{27x}{2} - \frac{\pi}{21})$:<br>$h'(x) = \frac{4}{9}(\cot(\frac{27x}{2} - \frac{\pi}{21}))' = \frac{4}{9}(-\frac{1}{\sin^2(\frac{27x}{2} - \frac{\pi}{21})}) \cdot (\frac{27x}{2})' = -\frac{4}{9}\frac{1}{\sin^2(\frac{27x}{2} - \frac{\pi}{21})} \cdot \frac{27}{2} = -\frac{4 \cdot 27}{9 \cdot 2 \cdot \sin^2(\frac{27x}{2} - \frac{\pi}{21})} = -\frac{6}{\sin^2(\frac{27x}{2} - \frac{\pi}{21})}$.<br><b>Ответ:</b> $h'(x) = -\frac{6}{\sin^2(\frac{27x}{2} - \frac{\pi}{21})}$.</p><p>Для функции $u(x) = -\frac{1}{11\pi}\cot(\frac{\pi}{11} - 22\pi x)$:<br>$u'(x) = -\frac{1}{11\pi}(\cot(\frac{\pi}{11} - 22\pi x))' = -\frac{1}{11\pi}(-\frac{1}{\sin^2(\frac{\pi}{11} - 22\pi x)}) \cdot (\frac{\pi}{11} - 22\pi x)' = \frac{1}{11\pi}\frac{1}{\sin^2(\frac{\pi}{11} - 22\pi x)} \cdot (-22\pi) = -\frac{2}{\sin^2(\frac{\pi}{11} - 22\pi x)}$.<br><b>Ответ:</b> $u'(x) = -\frac{2}{\sin^2(\frac{\pi}{11} - 22\pi x)}$.</p>" ] #original: array:6 [ "id" => 1551920 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1507} "task" => array:2 [ "refs" => "1120162" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Для функции $f(x) = \sin x$ производная является табличной.<br>$f'(x) = (\sin x)' = \cos x$.<br><b>Ответ:</b> $f'(x) = \cos x$.</p><p>Для функции $g(x) = \sin 2x$ применяем правило дифференцирования сложной функции: $(\sin(kx))' = k\cos(kx)$.<br>$g'(x) = (\sin 2x)' = \cos(2x) \cdot (2x)' = 2\cos 2x$.<br><b>Ответ:</b> $g'(x) = 2\cos 2x$.</p><p>Для функции $h(x) = \frac{1}{6}\sin(3x - \frac{\pi}{3})$ применяем правило дифференцирования сложной функции и вынесение константы за знак производной.<br>$h'(x) = \frac{1}{6}(\sin(3x - \frac{\pi}{3}))' = \frac{1}{6} \cos(3x - \frac{\pi}{3}) \cdot (3x - \frac{\pi}{3})' = \frac{1}{6} \cos(3x - \frac{\pi}{3}) \cdot 3 = \frac{3}{6}\cos(3x - \frac{\pi}{3}) = \frac{1}{2}\cos(3x - \frac{\pi}{3})$.<br><b>Ответ:</b> $h'(x) = \frac{1}{2}\cos(3x - \frac{\pi}{3})$.</p><p>Для функции $u(x) = -\frac{1}{\pi}\sin(\frac{\pi}{5} - 4\pi x)$ применяем те же правила.<br>$u'(x) = -\frac{1}{\pi}(\sin(\frac{\pi}{5} - 4\pi x))' = -\frac{1}{\pi}\cos(\frac{\pi}{5} - 4\pi x) \cdot (\frac{\pi}{5} - 4\pi x)' = -\frac{1}{\pi}\cos(\frac{\pi}{5} - 4\pi x) \cdot (-4\pi) = 4\cos(\frac{\pi}{5} - 4\pi x)$.<br><b>Ответ:</b> $u'(x) = 4\cos(\frac{\pi}{5} - 4\pi x)$.</p><p><strong>б)</strong></p><p>Для функции $f(x) = -\cos x$ производная равна:<br>$f'(x) = -(\cos x)' = -(-\sin x) = \sin x$.<br><b>Ответ:</b> $f'(x) = \sin x$.</p><p>Для функции $g(x) = \cos 3x$ применяем правило дифференцирования сложной функции: $(\cos(kx))' = -k\sin(kx)$.<br>$g'(x) = (\cos 3x)' = -\sin(3x) \cdot (3x)' = -3\sin 3x$.<br><b>Ответ:</b> $g'(x) = -3\sin 3x$.</p><p>Для функции $h(x) = \frac{1}{4}\cos(8x - \frac{\pi}{7})$:<br>$h'(x) = \frac{1}{4}(\cos(8x - \frac{\pi}{7}))' = \frac{1}{4}(-\sin(8x - \frac{\pi}{7})) \cdot (8x - \frac{\pi}{7})' = -\frac{1}{4}\sin(8x - \frac{\pi}{7}) \cdot 8 = -2\sin(8x - \frac{\pi}{7})$.<br><b>Ответ:</b> $h'(x) = -2\sin(8x - \frac{\pi}{7})$.</p><p>Для функции $u(x) = \frac{2}{\pi}\cos(\frac{\pi}{8} - \frac{3\pi x}{2})$:<br>$u'(x) = \frac{2}{\pi}(\cos(\frac{\pi}{8} - \frac{3\pi x}{2}))' = \frac{2}{\pi}(-\sin(\frac{\pi}{8} - \frac{3\pi x}{2})) \cdot (\frac{\pi}{8} - \frac{3\pi x}{2})' = -\frac{2}{\pi}\sin(\frac{\pi}{8} - \frac{3\pi x}{2}) \cdot (-\frac{3\pi}{2}) = 3\sin(\frac{\pi}{8} - \frac{3\pi x}{2})$.<br><b>Ответ:</b> $u'(x) = 3\sin(\frac{\pi}{8} - \frac{3\pi x}{2})$.</p><p><strong>в)</strong></p><p>Для функции $f(x) = 2\tan x$, используя табличную производную $(\tan x)' = \frac{1}{\cos^2 x}$:<br>$f'(x) = 2(\tan x)' = \frac{2}{\cos^2 x}$.<br><b>Ответ:</b> $f'(x) = \frac{2}{\cos^2 x}$.</p><p>Для функции $g(x) = \tan \frac{5x}{2}$, используя правило дифференцирования сложной функции:<br>$g'(x) = (\tan \frac{5x}{2})' = \frac{1}{\cos^2(\frac{5x}{2})} \cdot (\frac{5x}{2})' = \frac{1}{\cos^2(\frac{5x}{2})} \cdot \frac{5}{2} = \frac{5}{2\cos^2(\frac{5x}{2})}$.<br><b>Ответ:</b> $g'(x) = \frac{5}{2\cos^2(\frac{5x}{2})}$.</p><p>Для функции $h(x) = \frac{3}{7}\tan(\frac{14x}{3} - \frac{\pi}{9})$:<br>$h'(x) = \frac{3}{7}(\tan(\frac{14x}{3} - \frac{\pi}{9}))' = \frac{3}{7}\frac{1}{\cos^2(\frac{14x}{3} - \frac{\pi}{9})} \cdot (\frac{14x}{3} - \frac{\pi}{9})' = \frac{3}{7}\frac{1}{\cos^2(\frac{14x}{3} - \frac{\pi}{9})} \cdot \frac{14}{3} = \frac{2}{\cos^2(\frac{14x}{3} - \frac{\pi}{9})}$.<br><b>Ответ:</b> $h'(x) = \frac{2}{\cos^2(\frac{14x}{3} - \frac{\pi}{9})}$.</p><p>Для функции $u(x) = -\frac{1}{3}\tan(\frac{\pi}{5} - 9x)$:<br>$u'(x) = -\frac{1}{3}(\tan(\frac{\pi}{5} - 9x))' = -\frac{1}{3}\frac{1}{\cos^2(\frac{\pi}{5} - 9x)} \cdot (\frac{\pi}{5} - 9x)' = -\frac{1}{3}\frac{1}{\cos^2(\frac{\pi}{5} - 9x)} \cdot (-9) = \frac{3}{\cos^2(\frac{\pi}{5} - 9x)}$.<br><b>Ответ:</b> $u'(x) = \frac{3}{\cos^2(\frac{\pi}{5} - 9x)}$.</p><p><strong>г)</strong></p><p>Для функции $f(x) = -3\cot x$, используя табличную производную $(\cot x)' = -\frac{1}{\sin^2 x}$:<br>$f'(x) = -3(\cot x)' = -3(-\frac{1}{\sin^2 x}) = \frac{3}{\sin^2 x}$.<br><b>Ответ:</b> $f'(x) = \frac{3}{\sin^2 x}$.</p><p>Для функции $g(x) = 7\cot \frac{8x}{7}$, используя правило дифференцирования сложной функции:<br>$g'(x) = 7(\cot \frac{8x}{7})' = 7(-\frac{1}{\sin^2(\frac{8x}{7})}) \cdot (\frac{8x}{7})' = -7\frac{1}{\sin^2(\frac{8x}{7})} \cdot \frac{8}{7} = -\frac{8}{\sin^2(\frac{8x}{7})}$.<br><b>Ответ:</b> $g'(x) = -\frac{8}{\sin^2(\frac{8x}{7})}$.</p><p>Для функции $h(x) = \frac{4}{9}\cot(\frac{27x}{2} - \frac{\pi}{21})$:<br>$h'(x) = \frac{4}{9}(\cot(\frac{27x}{2} - \frac{\pi}{21}))' = \frac{4}{9}(-\frac{1}{\sin^2(\frac{27x}{2} - \frac{\pi}{21})}) \cdot (\frac{27x}{2})' = -\frac{4}{9}\frac{1}{\sin^2(\frac{27x}{2} - \frac{\pi}{21})} \cdot \frac{27}{2} = -\frac{4 \cdot 27}{9 \cdot 2 \cdot \sin^2(\frac{27x}{2} - \frac{\pi}{21})} = -\frac{6}{\sin^2(\frac{27x}{2} - \frac{\pi}{21})}$.<br><b>Ответ:</b> $h'(x) = -\frac{6}{\sin^2(\frac{27x}{2} - \frac{\pi}{21})}$.</p><p>Для функции $u(x) = -\frac{1}{11\pi}\cot(\frac{\pi}{11} - 22\pi x)$:<br>$u'(x) = -\frac{1}{11\pi}(\cot(\frac{\pi}{11} - 22\pi x))' = -\frac{1}{11\pi}(-\frac{1}{\sin^2(\frac{\pi}{11} - 22\pi x)}) \cdot (\frac{\pi}{11} - 22\pi x)' = \frac{1}{11\pi}\frac{1}{\sin^2(\frac{\pi}{11} - 22\pi x)} \cdot (-22\pi) = -\frac{2}{\sin^2(\frac{\pi}{11} - 22\pi x)}$.<br><b>Ответ:</b> $u'(x) = -\frac{2}{\sin^2(\frac{\pi}{11} - 22\pi x)}$.</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" => "1120163" "type" => "task" ] "previous" => array:2 [ "refs" => "1120161" "type" => 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№3 (с. 44)
Условие. №3 (с. 44)
скриншот условия
3. Найдите производные следующих функций:
а) $f(x)=\sin x$, $g(x)=\sin 2x$, $h(x)=\frac{1}{6}\sin\left(3x-\frac{\pi}{3}\right)$, $u(x)=-\frac{1}{\pi}\sin\left(\frac{\pi}{5}-4\pi x\right)$;
б) $f(x)=-\cos x$, $g(x)=\cos 3x$, $h(x)=\frac{1}{4}\cos\left(8x-\frac{\pi}{7}\right)$, $u(x)=-\frac{2}{\pi}\cos\left(\frac{\pi}{8}-\frac{3\pi x}{2}\right)$;
в) $f(x)=2\operatorname{tg}x$, $g(x)=\operatorname{tg}\frac{5x}{2}$, $h(x)=\frac{3}{7}\operatorname{tg}\left(\frac{14x}{3}-\frac{\pi}{9}\right)$, $u(x)=-\frac{1}{3}\operatorname{tg}\left(\frac{\pi}{5}-9x\right)$;
г) $f(x)=-3\operatorname{ctg}x$, $g(x)=7\operatorname{ctg}\frac{8x}{7}$, $h(x)=-\frac{4}{9}\operatorname{ctg}\left(\frac{27x}{2}-\frac{\pi}{21}\right)$, $u(x)=-\frac{1}{11\pi}\operatorname{ctg}\left(\frac{\pi}{11}-22\pi x\right)$.
Решение 2 (rus). №3 (с. 44)
а)
Для функции $f(x) = \sin x$ производная является табличной.
$f'(x) = (\sin x)' = \cos x$.
Ответ: $f'(x) = \cos x$.
Для функции $g(x) = \sin 2x$ применяем правило дифференцирования сложной функции: $(\sin(kx))' = k\cos(kx)$.
$g'(x) = (\sin 2x)' = \cos(2x) \cdot (2x)' = 2\cos 2x$.
Ответ: $g'(x) = 2\cos 2x$.
Для функции $h(x) = \frac{1}{6}\sin(3x - \frac{\pi}{3})$ применяем правило дифференцирования сложной функции и вынесение константы за знак производной.
$h'(x) = \frac{1}{6}(\sin(3x - \frac{\pi}{3}))' = \frac{1}{6} \cos(3x - \frac{\pi}{3}) \cdot (3x - \frac{\pi}{3})' = \frac{1}{6} \cos(3x - \frac{\pi}{3}) \cdot 3 = \frac{3}{6}\cos(3x - \frac{\pi}{3}) = \frac{1}{2}\cos(3x - \frac{\pi}{3})$.
Ответ: $h'(x) = \frac{1}{2}\cos(3x - \frac{\pi}{3})$.
Для функции $u(x) = -\frac{1}{\pi}\sin(\frac{\pi}{5} - 4\pi x)$ применяем те же правила.
$u'(x) = -\frac{1}{\pi}(\sin(\frac{\pi}{5} - 4\pi x))' = -\frac{1}{\pi}\cos(\frac{\pi}{5} - 4\pi x) \cdot (\frac{\pi}{5} - 4\pi x)' = -\frac{1}{\pi}\cos(\frac{\pi}{5} - 4\pi x) \cdot (-4\pi) = 4\cos(\frac{\pi}{5} - 4\pi x)$.
Ответ: $u'(x) = 4\cos(\frac{\pi}{5} - 4\pi x)$.
б)
Для функции $f(x) = -\cos x$ производная равна:
$f'(x) = -(\cos x)' = -(-\sin x) = \sin x$.
Ответ: $f'(x) = \sin x$.
Для функции $g(x) = \cos 3x$ применяем правило дифференцирования сложной функции: $(\cos(kx))' = -k\sin(kx)$.
$g'(x) = (\cos 3x)' = -\sin(3x) \cdot (3x)' = -3\sin 3x$.
Ответ: $g'(x) = -3\sin 3x$.
Для функции $h(x) = \frac{1}{4}\cos(8x - \frac{\pi}{7})$:
$h'(x) = \frac{1}{4}(\cos(8x - \frac{\pi}{7}))' = \frac{1}{4}(-\sin(8x - \frac{\pi}{7})) \cdot (8x - \frac{\pi}{7})' = -\frac{1}{4}\sin(8x - \frac{\pi}{7}) \cdot 8 = -2\sin(8x - \frac{\pi}{7})$.
Ответ: $h'(x) = -2\sin(8x - \frac{\pi}{7})$.
Для функции $u(x) = \frac{2}{\pi}\cos(\frac{\pi}{8} - \frac{3\pi x}{2})$:
$u'(x) = \frac{2}{\pi}(\cos(\frac{\pi}{8} - \frac{3\pi x}{2}))' = \frac{2}{\pi}(-\sin(\frac{\pi}{8} - \frac{3\pi x}{2})) \cdot (\frac{\pi}{8} - \frac{3\pi x}{2})' = -\frac{2}{\pi}\sin(\frac{\pi}{8} - \frac{3\pi x}{2}) \cdot (-\frac{3\pi}{2}) = 3\sin(\frac{\pi}{8} - \frac{3\pi x}{2})$.
Ответ: $u'(x) = 3\sin(\frac{\pi}{8} - \frac{3\pi x}{2})$.
в)
Для функции $f(x) = 2\tan x$, используя табличную производную $(\tan x)' = \frac{1}{\cos^2 x}$:
$f'(x) = 2(\tan x)' = \frac{2}{\cos^2 x}$.
Ответ: $f'(x) = \frac{2}{\cos^2 x}$.
Для функции $g(x) = \tan \frac{5x}{2}$, используя правило дифференцирования сложной функции:
$g'(x) = (\tan \frac{5x}{2})' = \frac{1}{\cos^2(\frac{5x}{2})} \cdot (\frac{5x}{2})' = \frac{1}{\cos^2(\frac{5x}{2})} \cdot \frac{5}{2} = \frac{5}{2\cos^2(\frac{5x}{2})}$.
Ответ: $g'(x) = \frac{5}{2\cos^2(\frac{5x}{2})}$.
Для функции $h(x) = \frac{3}{7}\tan(\frac{14x}{3} - \frac{\pi}{9})$:
$h'(x) = \frac{3}{7}(\tan(\frac{14x}{3} - \frac{\pi}{9}))' = \frac{3}{7}\frac{1}{\cos^2(\frac{14x}{3} - \frac{\pi}{9})} \cdot (\frac{14x}{3} - \frac{\pi}{9})' = \frac{3}{7}\frac{1}{\cos^2(\frac{14x}{3} - \frac{\pi}{9})} \cdot \frac{14}{3} = \frac{2}{\cos^2(\frac{14x}{3} - \frac{\pi}{9})}$.
Ответ: $h'(x) = \frac{2}{\cos^2(\frac{14x}{3} - \frac{\pi}{9})}$.
Для функции $u(x) = -\frac{1}{3}\tan(\frac{\pi}{5} - 9x)$:
$u'(x) = -\frac{1}{3}(\tan(\frac{\pi}{5} - 9x))' = -\frac{1}{3}\frac{1}{\cos^2(\frac{\pi}{5} - 9x)} \cdot (\frac{\pi}{5} - 9x)' = -\frac{1}{3}\frac{1}{\cos^2(\frac{\pi}{5} - 9x)} \cdot (-9) = \frac{3}{\cos^2(\frac{\pi}{5} - 9x)}$.
Ответ: $u'(x) = \frac{3}{\cos^2(\frac{\pi}{5} - 9x)}$.
г)
Для функции $f(x) = -3\cot x$, используя табличную производную $(\cot x)' = -\frac{1}{\sin^2 x}$:
$f'(x) = -3(\cot x)' = -3(-\frac{1}{\sin^2 x}) = \frac{3}{\sin^2 x}$.
Ответ: $f'(x) = \frac{3}{\sin^2 x}$.
Для функции $g(x) = 7\cot \frac{8x}{7}$, используя правило дифференцирования сложной функции:
$g'(x) = 7(\cot \frac{8x}{7})' = 7(-\frac{1}{\sin^2(\frac{8x}{7})}) \cdot (\frac{8x}{7})' = -7\frac{1}{\sin^2(\frac{8x}{7})} \cdot \frac{8}{7} = -\frac{8}{\sin^2(\frac{8x}{7})}$.
Ответ: $g'(x) = -\frac{8}{\sin^2(\frac{8x}{7})}$.
Для функции $h(x) = \frac{4}{9}\cot(\frac{27x}{2} - \frac{\pi}{21})$:
$h'(x) = \frac{4}{9}(\cot(\frac{27x}{2} - \frac{\pi}{21}))' = \frac{4}{9}(-\frac{1}{\sin^2(\frac{27x}{2} - \frac{\pi}{21})}) \cdot (\frac{27x}{2})' = -\frac{4}{9}\frac{1}{\sin^2(\frac{27x}{2} - \frac{\pi}{21})} \cdot \frac{27}{2} = -\frac{4 \cdot 27}{9 \cdot 2 \cdot \sin^2(\frac{27x}{2} - \frac{\pi}{21})} = -\frac{6}{\sin^2(\frac{27x}{2} - \frac{\pi}{21})}$.
Ответ: $h'(x) = -\frac{6}{\sin^2(\frac{27x}{2} - \frac{\pi}{21})}$.
Для функции $u(x) = -\frac{1}{11\pi}\cot(\frac{\pi}{11} - 22\pi x)$:
$u'(x) = -\frac{1}{11\pi}(\cot(\frac{\pi}{11} - 22\pi x))' = -\frac{1}{11\pi}(-\frac{1}{\sin^2(\frac{\pi}{11} - 22\pi x)}) \cdot (\frac{\pi}{11} - 22\pi x)' = \frac{1}{11\pi}\frac{1}{\sin^2(\frac{\pi}{11} - 22\pi x)} \cdot (-22\pi) = -\frac{2}{\sin^2(\frac{\pi}{11} - 22\pi x)}$.
Ответ: $u'(x) = -\frac{2}{\sin^2(\frac{\pi}{11} - 22\pi x)}$.
Другие задания:
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