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Номер 28, страница 48, 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 28, страница 48.
App\Models\Task {#1024 // 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" => 1120187 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "48" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/5-2-5zad-28" "field_display_title" => "28" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #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 {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #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 {#333 #items: array:1 [ 0 => App\Models\Term {#1033 #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 {#1030 #items: array:1 [ 0 => App\Models\Book {#1032 #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 {#1034 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1036 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1038 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1040 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1056 …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 {#1057 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1060 …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 {#1034 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1036 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1038 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1040 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1056 …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 {#1057 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1060 …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 {#1061 #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 {#333} "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 {#1030} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1061} ] #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 {#1062 #items: array:5 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1063 #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 {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #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 {#1066 #items: array:1 [ 0 => App\Models\Book {#1067 #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 {#1068 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1090 …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 {#1091 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1094 …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 {#1068 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1090 …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 {#1091 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1094 …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 {#1095 #items: array:1 [ 0 => App\Models\Branch {#1096 #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 {#1097 …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 {#1099 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1100 …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 {#1097 …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 {#1099 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1100 …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 {#1064} "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 {#1066} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1095} ] #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 {#1101 #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 {#1102 #items: array:1 [ 0 => App\Models\Term {#1103 #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 {#1104 #items: array:1 [ 0 => App\Models\Book {#1105 #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 {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …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 {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …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 {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …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 {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …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 {#1133 #items: array:1 [ 0 => App\Models\Branch {#1134 #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 {#1135 …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 {#1137 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1138 …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 {#1135 …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 {#1137 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1138 …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 {#1102} "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 {#1104} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1133} ] #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 {#1172 #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 {#1173 #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 {#1174 #items: array:1 [ 0 => App\Models\Book {#1175 #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 {#1176 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1195 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1196 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1197 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1198 …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 {#1199 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1201 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1202 …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 {#1176 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1195 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1196 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1197 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1198 …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 {#1199 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1201 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1202 …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 {#1203 #items: array:1 [ 0 => App\Models\Branch {#1204 #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 {#1205 …2} …17 ] #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 {#1173} "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 {#1174} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1203} ] #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 {#1275 #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 {#1276 #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 {#1277 #items: array:1 [ 0 => App\Models\Book {#1278 #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 {#1306 #items: array:1 [ 0 => App\Models\Branch {#1307 #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 {#1276} "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 {#1277} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1306} ] #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 {#1382 #items: array:2 [ 0 => App\Models\Element {#1383 #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" => 1300909 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1384 #items: array:1 [ 0 => App\Models\Edition {#1385 #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" => "1120187" "type" => "task" ] "text" => "<p><strong>28.</strong> При каких значениях переменной значение функции $g(x)$ равно значению производной функции $f(x)$, если:</p><p><strong>а)</strong> $f(x) = -\frac{2\cos5x}{5} + \cos\frac{\pi}{3}$, $g(x) = \sin3x + \sin7x$;</p><p><strong>б)</strong> $f(x) = 2x + \cos^2\frac{4\pi}{17}$, $g(x) = \sin^2 2x + \sin^2 3x + \sin^2 4x + \sin^2 5x$;</p><p><strong>в)</strong> $f(x) = \frac{\cos8x}{8} - \frac{\cos2x}{2}$, $g(x) = 2\sin3x \cos x$;</p><p><strong>г)</strong> $f(x) = \frac{\sin4x}{8} + \frac{3}{4}x + \frac{\sin2\pi}{21}$, $g(x) = \sin^4 x + \cos^4 x?$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "28-1.jpg" "alt" => null "width" => "1707" "height" => 821 "path" => "/media/algebra_10/baspasy-u19/0-5-2-5zad/28-1.webp?ts=1753534834" ] ] ] #original: array:7 [ "id" => 1300909 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1384} "task" => array:2 [ "refs" => "1120187" "type" => "task" ] "text" => "<p><strong>28.</strong> При каких значениях переменной значение функции $g(x)$ равно значению производной функции $f(x)$, если:</p><p><strong>а)</strong> $f(x) = -\frac{2\cos5x}{5} + \cos\frac{\pi}{3}$, $g(x) = \sin3x + \sin7x$;</p><p><strong>б)</strong> $f(x) = 2x + \cos^2\frac{4\pi}{17}$, $g(x) = \sin^2 2x + \sin^2 3x + \sin^2 4x + \sin^2 5x$;</p><p><strong>в)</strong> $f(x) = \frac{\cos8x}{8} - \frac{\cos2x}{2}$, $g(x) = 2\sin3x \cos x$;</p><p><strong>г)</strong> $f(x) = \frac{\sin4x}{8} + \frac{3}{4}x + \frac{\sin2\pi}{21}$, $g(x) = \sin^4 x + \cos^4 x?$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "28-1.jpg" "alt" => null "width" => "1707" "height" => 821 "path" => "/media/algebra_10/baspasy-u19/0-5-2-5zad/28-1.webp?ts=1753534834" ] ] ] #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 {#1391 #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" => 1551945 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1392 #items: array:1 [ 0 => App\Models\Edition {#1393 #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" => "1120187" "type" => "task" ] "text" => "<p><strong>а)</strong></p> <p>Сначала найдем производную функции $f(x) = -\frac{2\cos{5x}}{5} + \cos{\frac{\pi}{3}}$.</p> <p>Поскольку $\cos{\frac{\pi}{3}}$ является константой, ее производная равна нулю. Производная от $\cos{5x}$ равна $-5\sin{5x}$.</p> <p>$f'(x) = \left(-\frac{2}{5}\cos{5x} + \cos{\frac{\pi}{3}}\right)' = -\frac{2}{5}(-\sin{5x} \cdot 5) + 0 = 2\sin{5x}$.</p> <p>Теперь приравняем значение функции $g(x)$ к значению производной $f'(x)$:</p> <p>$g(x) = f'(x)$</p> <p>$\sin{3x} + \sin{7x} = 2\sin{5x}$</p> <p>Используем формулу суммы синусов: $\sin{\alpha} + \sin{\beta} = 2\sin{\frac{\alpha+\beta}{2}}\cos{\frac{\alpha-\beta}{2}}$.</p> <p>$2\sin{\frac{3x+7x}{2}}\cos{\frac{7x-3x}{2}} = 2\sin{5x}$</p> <p>$2\sin{5x}\cos{2x} = 2\sin{5x}$</p> <p>$2\sin{5x}\cos{2x} - 2\sin{5x} = 0$</p> <p>$2\sin{5x}(\cos{2x} - 1) = 0$</p> <p>Это уравнение распадается на два:</p> <p>1) $\sin{5x} = 0$</p> <p>$5x = \pi k, \quad k \in Z$</p> <p>$x = \frac{\pi k}{5}, \quad k \in Z$</p> <p>2) $\cos{2x} - 1 = 0 \implies \cos{2x} = 1$</p> <p>$2x = 2\pi n, \quad n \in Z$</p> <p>$x = \pi n, \quad n \in Z$</p> <p>Заметим, что множество решений $x = \pi n$ является подмножеством множества решений $x = \frac{\pi k}{5}$ (при $k=5n$). Следовательно, общее решение - это $x = \frac{\pi k}{5}$.</p> <p><strong>Ответ:</strong> $x = \frac{\pi k}{5}, \quad k \in Z$.</p> <p><strong>б)</strong></p> <p>Найдем производную функции $f(x) = 2x + \cos^2{\frac{4\pi}{17}}$.</p> <p>Слагаемое $\cos^2{\frac{4\pi}{17}}$ является константой, поэтому его производная равна нулю.</p> <p>$f'(x) = (2x)' + (\cos^2{\frac{4\pi}{17}})' = 2 + 0 = 2$.</p> <p>Приравниваем $g(x)$ к $f'(x)$:</p> <p>$\sin^2{2x} + \sin^2{3x} + \sin^2{4x} + \sin^2{5x} = 2$</p> <p>Используем формулу понижения степени $\sin^2{\alpha} = \frac{1-\cos{2\alpha}}{2}$:</p> <p>$\frac{1-\cos{4x}}{2} + \frac{1-\cos{6x}}{2} + \frac{1-\cos{8x}}{2} + \frac{1-\cos{10x}}{2} = 2$</p> <p>$1-\cos{4x} + 1-\cos{6x} + 1-\cos{8x} + 1-\cos{10x} = 4$</p> <p>$4 - (\cos{4x} + \cos{6x} + \cos{8x} + \cos{10x}) = 4$</p> <p>$\cos{4x} + \cos{6x} + \cos{8x} + \cos{10x} = 0$</p> <p>Сгруппируем слагаемые и применим формулу суммы косинусов $\cos{\alpha} + \cos{\beta} = 2\cos{\frac{\alpha+\beta}{2}}\cos{\frac{\alpha-\beta}{2}}$:</p> <p>$(\cos{10x} + \cos{4x}) + (\cos{8x} + \cos{6x}) = 0$</p> <p>$2\cos{\frac{10x+4x}{2}}\cos{\frac{10x-4x}{2}} + 2\cos{\frac{8x+6x}{2}}\cos{\frac{8x-6x}{2}} = 0$</p> <p>$2\cos{7x}\cos{3x} + 2\cos{7x}\cos{x} = 0$</p> <p>$2\cos{7x}(\cos{3x} + \cos{x}) = 0$</p> <p>Снова применяем формулу суммы косинусов:</p> <p>$2\cos{7x}(2\cos{\frac{3x+x}{2}}\cos{\frac{3x-x}{2}}) = 0$</p> <p>$4\cos{7x}\cos{2x}\cos{x} = 0$</p> <p>Получаем три уравнения:</p> <p>1) $\cos{x} = 0 \implies x = \frac{\pi}{2} + \pi k, \quad k \in Z$</p> <p>2) $\cos{2x} = 0 \implies 2x = \frac{\pi}{2} + \pi n \implies x = \frac{\pi}{4} + \frac{\pi n}{2}, \quad n \in Z$</p> <p>3) $\cos{7x} = 0 \implies 7x = \frac{\pi}{2} + \pi m \implies x = \frac{\pi}{14} + \frac{\pi m}{7}, \quad m \in Z$</p> <p>Решения $x = \frac{\pi}{2} + \pi k$ являются подмножеством решений $x = \frac{\pi}{14} + \frac{\pi m}{7}$ (при $m = 3+7k$), поэтому первые можно не указывать в ответе.</p> <p><strong>Ответ:</strong> $x = \frac{\pi}{4} + \frac{\pi n}{2}, \quad n \in Z; \quad x = \frac{\pi}{14} + \frac{\pi m}{7}, \quad m \in Z$.</p> <p><strong>в)</strong></p> <p>Найдем производную функции $f(x) = -\frac{\cos{8x}}{8} - \frac{\cos{2x}}{2}$.</p> <p>$f'(x) = \left(-\frac{1}{8}\cos{8x}\right)' - \left(\frac{1}{2}\cos{2x}\right)' = -\frac{1}{8}(-\sin{8x} \cdot 8) - \frac{1}{2}(-\sin{2x} \cdot 2) = \sin{8x} + \sin{2x}$.</p> <p>Преобразуем функцию $g(x)$ по формуле произведения синуса на косинус $2\sin{\alpha}\cos{\beta} = \sin{(\alpha+\beta)} + \sin{(\alpha-\beta)}$:</p> <p>$g(x) = 2\sin{3x}\cos{x} = \sin(3x+x) + \sin(3x-x) = \sin{4x} + \sin{2x}$.</p> <p>Приравняем $g(x)$ к $f'(x)$:</p> <p>$\sin{4x} + \sin{2x} = \sin{8x} + \sin{2x}$</p> <p>$\sin{4x} = \sin{8x}$</p> <p>$\sin{8x} - \sin{4x} = 0$</p> <p>Используем формулу разности синусов $\sin{\alpha} - \sin{\beta} = 2\cos{\frac{\alpha+\beta}{2}}\sin{\frac{\alpha-\beta}{2}}$:</p> <p>$2\cos{\frac{8x+4x}{2}}\sin{\frac{8x-4x}{2}} = 0$</p> <p>$2\cos{6x}\sin{2x} = 0$</p> <p>Получаем два уравнения:</p> <p>1) $\sin{2x} = 0$</p> <p>$2x = \pi n \implies x = \frac{\pi n}{2}, \quad n \in Z$</p> <p>2) $\cos{6x} = 0$</p> <p>$6x = \frac{\pi}{2} + \pi k \implies x = \frac{\pi}{12} + \frac{\pi k}{6}, \quad k \in Z$</p> <p><strong>Ответ:</strong> $x = \frac{\pi n}{2}, \quad n \in Z; \quad x = \frac{\pi}{12} + \frac{\pi k}{6}, \quad k \in Z$.</p> <p><strong>г)</strong></p> <p>Найдем производную функции $f(x) = \frac{\sin{4x}}{8} + \frac{3}{4}x + \frac{\sin{2\pi}}{21}$.</p> <p>Слагаемое $\frac{\sin{2\pi}}{21}$ является константой (равно 0), его производная равна нулю.</p> <p>$f'(x) = \left(\frac{1}{8}\sin{4x}\right)' + \left(\frac{3}{4}x\right)' + 0 = \frac{1}{8}(\cos{4x} \cdot 4) + \frac{3}{4} = \frac{1}{2}\cos{4x} + \frac{3}{4}$.</p> <p>Теперь преобразуем функцию $g(x) = \sin^4{x} + \cos^4{x}$:</p> <p>$g(x) = (\sin^2{x} + \cos^2{x})^2 - 2\sin^2{x}\cos^2{x} = 1^2 - 2(\sin{x}\cos{x})^2 = 1 - 2\left(\frac{\sin{2x}}{2}\right)^2 = 1 - 2\frac{\sin^2{2x}}{4} = 1 - \frac{1}{2}\sin^2{2x}$.</p> <p>Используем формулу понижения степени $\sin^2{\alpha} = \frac{1-\cos{2\alpha}}{2}$:</p> <p>$g(x) = 1 - \frac{1}{2}\left(\frac{1-\cos{4x}}{2}\right) = 1 - \frac{1-\cos{4x}}{4} = \frac{4 - (1-\cos{4x})}{4} = \frac{3+\cos{4x}}{4} = \frac{3}{4} + \frac{1}{4}\cos{4x}$.</p> <p>Приравняем $g(x)$ к $f'(x)$:</p> <p>$\frac{3}{4} + \frac{1}{4}\cos{4x} = \frac{1}{2}\cos{4x} + \frac{3}{4}$</p> <p>$\frac{1}{4}\cos{4x} = \frac{1}{2}\cos{4x}$</p> <p>$\frac{1}{2}\cos{4x} - \frac{1}{4}\cos{4x} = 0$</p> <p>$\frac{1}{4}\cos{4x} = 0$</p> <p>$\cos{4x} = 0$</p> <p>$4x = \frac{\pi}{2} + \pi k, \quad k \in Z$</p> <p>$x = \frac{\pi}{8} + \frac{\pi k}{4}, \quad k \in Z$</p> <p><strong>Ответ:</strong> $x = \frac{\pi}{8} + \frac{\pi k}{4}, \quad k \in Z$.</p>" ] #original: array:6 [ "id" => 1551945 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1392} "task" => array:2 [ "refs" => "1120187" "type" => "task" ] "text" => "<p><strong>а)</strong></p> <p>Сначала найдем производную функции $f(x) = -\frac{2\cos{5x}}{5} + \cos{\frac{\pi}{3}}$.</p> <p>Поскольку $\cos{\frac{\pi}{3}}$ является константой, ее производная равна нулю. Производная от $\cos{5x}$ равна $-5\sin{5x}$.</p> <p>$f'(x) = \left(-\frac{2}{5}\cos{5x} + \cos{\frac{\pi}{3}}\right)' = -\frac{2}{5}(-\sin{5x} \cdot 5) + 0 = 2\sin{5x}$.</p> <p>Теперь приравняем значение функции $g(x)$ к значению производной $f'(x)$:</p> <p>$g(x) = f'(x)$</p> <p>$\sin{3x} + \sin{7x} = 2\sin{5x}$</p> <p>Используем формулу суммы синусов: $\sin{\alpha} + \sin{\beta} = 2\sin{\frac{\alpha+\beta}{2}}\cos{\frac{\alpha-\beta}{2}}$.</p> <p>$2\sin{\frac{3x+7x}{2}}\cos{\frac{7x-3x}{2}} = 2\sin{5x}$</p> <p>$2\sin{5x}\cos{2x} = 2\sin{5x}$</p> <p>$2\sin{5x}\cos{2x} - 2\sin{5x} = 0$</p> <p>$2\sin{5x}(\cos{2x} - 1) = 0$</p> <p>Это уравнение распадается на два:</p> <p>1) $\sin{5x} = 0$</p> <p>$5x = \pi k, \quad k \in Z$</p> <p>$x = \frac{\pi k}{5}, \quad k \in Z$</p> <p>2) $\cos{2x} - 1 = 0 \implies \cos{2x} = 1$</p> <p>$2x = 2\pi n, \quad n \in Z$</p> <p>$x = \pi n, \quad n \in Z$</p> <p>Заметим, что множество решений $x = \pi n$ является подмножеством множества решений $x = \frac{\pi k}{5}$ (при $k=5n$). Следовательно, общее решение - это $x = \frac{\pi k}{5}$.</p> <p><strong>Ответ:</strong> $x = \frac{\pi k}{5}, \quad k \in Z$.</p> <p><strong>б)</strong></p> <p>Найдем производную функции $f(x) = 2x + \cos^2{\frac{4\pi}{17}}$.</p> <p>Слагаемое $\cos^2{\frac{4\pi}{17}}$ является константой, поэтому его производная равна нулю.</p> <p>$f'(x) = (2x)' + (\cos^2{\frac{4\pi}{17}})' = 2 + 0 = 2$.</p> <p>Приравниваем $g(x)$ к $f'(x)$:</p> <p>$\sin^2{2x} + \sin^2{3x} + \sin^2{4x} + \sin^2{5x} = 2$</p> <p>Используем формулу понижения степени $\sin^2{\alpha} = \frac{1-\cos{2\alpha}}{2}$:</p> <p>$\frac{1-\cos{4x}}{2} + \frac{1-\cos{6x}}{2} + \frac{1-\cos{8x}}{2} + \frac{1-\cos{10x}}{2} = 2$</p> <p>$1-\cos{4x} + 1-\cos{6x} + 1-\cos{8x} + 1-\cos{10x} = 4$</p> <p>$4 - (\cos{4x} + \cos{6x} + \cos{8x} + \cos{10x}) = 4$</p> <p>$\cos{4x} + \cos{6x} + \cos{8x} + \cos{10x} = 0$</p> <p>Сгруппируем слагаемые и применим формулу суммы косинусов $\cos{\alpha} + \cos{\beta} = 2\cos{\frac{\alpha+\beta}{2}}\cos{\frac{\alpha-\beta}{2}}$:</p> <p>$(\cos{10x} + \cos{4x}) + (\cos{8x} + \cos{6x}) = 0$</p> <p>$2\cos{\frac{10x+4x}{2}}\cos{\frac{10x-4x}{2}} + 2\cos{\frac{8x+6x}{2}}\cos{\frac{8x-6x}{2}} = 0$</p> <p>$2\cos{7x}\cos{3x} + 2\cos{7x}\cos{x} = 0$</p> <p>$2\cos{7x}(\cos{3x} + \cos{x}) = 0$</p> <p>Снова применяем формулу суммы косинусов:</p> <p>$2\cos{7x}(2\cos{\frac{3x+x}{2}}\cos{\frac{3x-x}{2}}) = 0$</p> <p>$4\cos{7x}\cos{2x}\cos{x} = 0$</p> <p>Получаем три уравнения:</p> <p>1) $\cos{x} = 0 \implies x = \frac{\pi}{2} + \pi k, \quad k \in Z$</p> <p>2) $\cos{2x} = 0 \implies 2x = \frac{\pi}{2} + \pi n \implies x = \frac{\pi}{4} + \frac{\pi n}{2}, \quad n \in Z$</p> <p>3) $\cos{7x} = 0 \implies 7x = \frac{\pi}{2} + \pi m \implies x = \frac{\pi}{14} + \frac{\pi m}{7}, \quad m \in Z$</p> <p>Решения $x = \frac{\pi}{2} + \pi k$ являются подмножеством решений $x = \frac{\pi}{14} + \frac{\pi m}{7}$ (при $m = 3+7k$), поэтому первые можно не указывать в ответе.</p> <p><strong>Ответ:</strong> $x = \frac{\pi}{4} + \frac{\pi n}{2}, \quad n \in Z; \quad x = \frac{\pi}{14} + \frac{\pi m}{7}, \quad m \in Z$.</p> <p><strong>в)</strong></p> <p>Найдем производную функции $f(x) = -\frac{\cos{8x}}{8} - \frac{\cos{2x}}{2}$.</p> <p>$f'(x) = \left(-\frac{1}{8}\cos{8x}\right)' - \left(\frac{1}{2}\cos{2x}\right)' = -\frac{1}{8}(-\sin{8x} \cdot 8) - \frac{1}{2}(-\sin{2x} \cdot 2) = \sin{8x} + \sin{2x}$.</p> <p>Преобразуем функцию $g(x)$ по формуле произведения синуса на косинус $2\sin{\alpha}\cos{\beta} = \sin{(\alpha+\beta)} + \sin{(\alpha-\beta)}$:</p> <p>$g(x) = 2\sin{3x}\cos{x} = \sin(3x+x) + \sin(3x-x) = \sin{4x} + \sin{2x}$.</p> <p>Приравняем $g(x)$ к $f'(x)$:</p> <p>$\sin{4x} + \sin{2x} = \sin{8x} + \sin{2x}$</p> <p>$\sin{4x} = \sin{8x}$</p> <p>$\sin{8x} - \sin{4x} = 0$</p> <p>Используем формулу разности синусов $\sin{\alpha} - \sin{\beta} = 2\cos{\frac{\alpha+\beta}{2}}\sin{\frac{\alpha-\beta}{2}}$:</p> <p>$2\cos{\frac{8x+4x}{2}}\sin{\frac{8x-4x}{2}} = 0$</p> <p>$2\cos{6x}\sin{2x} = 0$</p> <p>Получаем два уравнения:</p> <p>1) $\sin{2x} = 0$</p> <p>$2x = \pi n \implies x = \frac{\pi n}{2}, \quad n \in Z$</p> <p>2) $\cos{6x} = 0$</p> <p>$6x = \frac{\pi}{2} + \pi k \implies x = \frac{\pi}{12} + \frac{\pi k}{6}, \quad k \in Z$</p> <p><strong>Ответ:</strong> $x = \frac{\pi n}{2}, \quad n \in Z; \quad x = \frac{\pi}{12} + \frac{\pi k}{6}, \quad k \in Z$.</p> <p><strong>г)</strong></p> <p>Найдем производную функции $f(x) = \frac{\sin{4x}}{8} + \frac{3}{4}x + \frac{\sin{2\pi}}{21}$.</p> <p>Слагаемое $\frac{\sin{2\pi}}{21}$ является константой (равно 0), его производная равна нулю.</p> <p>$f'(x) = \left(\frac{1}{8}\sin{4x}\right)' + \left(\frac{3}{4}x\right)' + 0 = \frac{1}{8}(\cos{4x} \cdot 4) + \frac{3}{4} = \frac{1}{2}\cos{4x} + \frac{3}{4}$.</p> <p>Теперь преобразуем функцию $g(x) = \sin^4{x} + \cos^4{x}$:</p> <p>$g(x) = (\sin^2{x} + \cos^2{x})^2 - 2\sin^2{x}\cos^2{x} = 1^2 - 2(\sin{x}\cos{x})^2 = 1 - 2\left(\frac{\sin{2x}}{2}\right)^2 = 1 - 2\frac{\sin^2{2x}}{4} = 1 - \frac{1}{2}\sin^2{2x}$.</p> <p>Используем формулу понижения степени $\sin^2{\alpha} = \frac{1-\cos{2\alpha}}{2}$:</p> <p>$g(x) = 1 - \frac{1}{2}\left(\frac{1-\cos{4x}}{2}\right) = 1 - \frac{1-\cos{4x}}{4} = \frac{4 - (1-\cos{4x})}{4} = \frac{3+\cos{4x}}{4} = \frac{3}{4} + \frac{1}{4}\cos{4x}$.</p> <p>Приравняем $g(x)$ к $f'(x)$:</p> <p>$\frac{3}{4} + \frac{1}{4}\cos{4x} = \frac{1}{2}\cos{4x} + \frac{3}{4}$</p> <p>$\frac{1}{4}\cos{4x} = \frac{1}{2}\cos{4x}$</p> <p>$\frac{1}{2}\cos{4x} - \frac{1}{4}\cos{4x} = 0$</p> <p>$\frac{1}{4}\cos{4x} = 0$</p> <p>$\cos{4x} = 0$</p> <p>$4x = \frac{\pi}{2} + \pi k, \quad k \in Z$</p> <p>$x = \frac{\pi}{8} + \frac{\pi k}{4}, \quad k \in Z$</p> <p><strong>Ответ:</strong> $x = \frac{\pi}{8} + \frac{\pi k}{4}, \quad k \in Z$.</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" => "1120188" "type" => "task" ] "previous" => array:2 [ "refs" => "1120186" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1399 #items: array:1 [ 0 => App\Models\Book {#1400 #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 {#1401 #items: array:1 [ 0 => App\Models\Term {#1402 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1403 #items: array:1 [ 0 => App\Models\Term {#1404 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1405 #items: array:1 [ 0 => App\Models\Term {#1406 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1407 #items: array:3 [ 0 => App\Models\Term {#1408 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1409 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1410 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1411 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1412 #items: array:1 [ 0 => App\Models\Term {#1413 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1414 #items: array:1 [ 0 => App\Models\Term {#1415 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1416 #items: array:1 [ 0 => App\Models\Term {#1417 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1418 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1419 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1420 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1421 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1422 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1423 #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 {#1424 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1425 #items: 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[] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№28 (с. 48)
Условие. №28 (с. 48)
скриншот условия
28. При каких значениях переменной значение функции $g(x)$ равно значению производной функции $f(x)$, если:
а) $f(x) = -\frac{2\cos5x}{5} + \cos\frac{\pi}{3}$, $g(x) = \sin3x + \sin7x$;
б) $f(x) = 2x + \cos^2\frac{4\pi}{17}$, $g(x) = \sin^2 2x + \sin^2 3x + \sin^2 4x + \sin^2 5x$;
в) $f(x) = \frac{\cos8x}{8} - \frac{\cos2x}{2}$, $g(x) = 2\sin3x \cos x$;
г) $f(x) = \frac{\sin4x}{8} + \frac{3}{4}x + \frac{\sin2\pi}{21}$, $g(x) = \sin^4 x + \cos^4 x?$
Решение 2 (rus). №28 (с. 48)
а)
Сначала найдем производную функции $f(x) = -\frac{2\cos{5x}}{5} + \cos{\frac{\pi}{3}}$.
Поскольку $\cos{\frac{\pi}{3}}$ является константой, ее производная равна нулю. Производная от $\cos{5x}$ равна $-5\sin{5x}$.
$f'(x) = \left(-\frac{2}{5}\cos{5x} + \cos{\frac{\pi}{3}}\right)' = -\frac{2}{5}(-\sin{5x} \cdot 5) + 0 = 2\sin{5x}$.
Теперь приравняем значение функции $g(x)$ к значению производной $f'(x)$:
$g(x) = f'(x)$
$\sin{3x} + \sin{7x} = 2\sin{5x}$
Используем формулу суммы синусов: $\sin{\alpha} + \sin{\beta} = 2\sin{\frac{\alpha+\beta}{2}}\cos{\frac{\alpha-\beta}{2}}$.
$2\sin{\frac{3x+7x}{2}}\cos{\frac{7x-3x}{2}} = 2\sin{5x}$
$2\sin{5x}\cos{2x} = 2\sin{5x}$
$2\sin{5x}\cos{2x} - 2\sin{5x} = 0$
$2\sin{5x}(\cos{2x} - 1) = 0$
Это уравнение распадается на два:
1) $\sin{5x} = 0$
$5x = \pi k, \quad k \in Z$
$x = \frac{\pi k}{5}, \quad k \in Z$
2) $\cos{2x} - 1 = 0 \implies \cos{2x} = 1$
$2x = 2\pi n, \quad n \in Z$
$x = \pi n, \quad n \in Z$
Заметим, что множество решений $x = \pi n$ является подмножеством множества решений $x = \frac{\pi k}{5}$ (при $k=5n$). Следовательно, общее решение - это $x = \frac{\pi k}{5}$.
Ответ: $x = \frac{\pi k}{5}, \quad k \in Z$.
б)
Найдем производную функции $f(x) = 2x + \cos^2{\frac{4\pi}{17}}$.
Слагаемое $\cos^2{\frac{4\pi}{17}}$ является константой, поэтому его производная равна нулю.
$f'(x) = (2x)' + (\cos^2{\frac{4\pi}{17}})' = 2 + 0 = 2$.
Приравниваем $g(x)$ к $f'(x)$:
$\sin^2{2x} + \sin^2{3x} + \sin^2{4x} + \sin^2{5x} = 2$
Используем формулу понижения степени $\sin^2{\alpha} = \frac{1-\cos{2\alpha}}{2}$:
$\frac{1-\cos{4x}}{2} + \frac{1-\cos{6x}}{2} + \frac{1-\cos{8x}}{2} + \frac{1-\cos{10x}}{2} = 2$
$1-\cos{4x} + 1-\cos{6x} + 1-\cos{8x} + 1-\cos{10x} = 4$
$4 - (\cos{4x} + \cos{6x} + \cos{8x} + \cos{10x}) = 4$
$\cos{4x} + \cos{6x} + \cos{8x} + \cos{10x} = 0$
Сгруппируем слагаемые и применим формулу суммы косинусов $\cos{\alpha} + \cos{\beta} = 2\cos{\frac{\alpha+\beta}{2}}\cos{\frac{\alpha-\beta}{2}}$:
$(\cos{10x} + \cos{4x}) + (\cos{8x} + \cos{6x}) = 0$
$2\cos{\frac{10x+4x}{2}}\cos{\frac{10x-4x}{2}} + 2\cos{\frac{8x+6x}{2}}\cos{\frac{8x-6x}{2}} = 0$
$2\cos{7x}\cos{3x} + 2\cos{7x}\cos{x} = 0$
$2\cos{7x}(\cos{3x} + \cos{x}) = 0$
Снова применяем формулу суммы косинусов:
$2\cos{7x}(2\cos{\frac{3x+x}{2}}\cos{\frac{3x-x}{2}}) = 0$
$4\cos{7x}\cos{2x}\cos{x} = 0$
Получаем три уравнения:
1) $\cos{x} = 0 \implies x = \frac{\pi}{2} + \pi k, \quad k \in Z$
2) $\cos{2x} = 0 \implies 2x = \frac{\pi}{2} + \pi n \implies x = \frac{\pi}{4} + \frac{\pi n}{2}, \quad n \in Z$
3) $\cos{7x} = 0 \implies 7x = \frac{\pi}{2} + \pi m \implies x = \frac{\pi}{14} + \frac{\pi m}{7}, \quad m \in Z$
Решения $x = \frac{\pi}{2} + \pi k$ являются подмножеством решений $x = \frac{\pi}{14} + \frac{\pi m}{7}$ (при $m = 3+7k$), поэтому первые можно не указывать в ответе.
Ответ: $x = \frac{\pi}{4} + \frac{\pi n}{2}, \quad n \in Z; \quad x = \frac{\pi}{14} + \frac{\pi m}{7}, \quad m \in Z$.
в)
Найдем производную функции $f(x) = -\frac{\cos{8x}}{8} - \frac{\cos{2x}}{2}$.
$f'(x) = \left(-\frac{1}{8}\cos{8x}\right)' - \left(\frac{1}{2}\cos{2x}\right)' = -\frac{1}{8}(-\sin{8x} \cdot 8) - \frac{1}{2}(-\sin{2x} \cdot 2) = \sin{8x} + \sin{2x}$.
Преобразуем функцию $g(x)$ по формуле произведения синуса на косинус $2\sin{\alpha}\cos{\beta} = \sin{(\alpha+\beta)} + \sin{(\alpha-\beta)}$:
$g(x) = 2\sin{3x}\cos{x} = \sin(3x+x) + \sin(3x-x) = \sin{4x} + \sin{2x}$.
Приравняем $g(x)$ к $f'(x)$:
$\sin{4x} + \sin{2x} = \sin{8x} + \sin{2x}$
$\sin{4x} = \sin{8x}$
$\sin{8x} - \sin{4x} = 0$
Используем формулу разности синусов $\sin{\alpha} - \sin{\beta} = 2\cos{\frac{\alpha+\beta}{2}}\sin{\frac{\alpha-\beta}{2}}$:
$2\cos{\frac{8x+4x}{2}}\sin{\frac{8x-4x}{2}} = 0$
$2\cos{6x}\sin{2x} = 0$
Получаем два уравнения:
1) $\sin{2x} = 0$
$2x = \pi n \implies x = \frac{\pi n}{2}, \quad n \in Z$
2) $\cos{6x} = 0$
$6x = \frac{\pi}{2} + \pi k \implies x = \frac{\pi}{12} + \frac{\pi k}{6}, \quad k \in Z$
Ответ: $x = \frac{\pi n}{2}, \quad n \in Z; \quad x = \frac{\pi}{12} + \frac{\pi k}{6}, \quad k \in Z$.
г)
Найдем производную функции $f(x) = \frac{\sin{4x}}{8} + \frac{3}{4}x + \frac{\sin{2\pi}}{21}$.
Слагаемое $\frac{\sin{2\pi}}{21}$ является константой (равно 0), его производная равна нулю.
$f'(x) = \left(\frac{1}{8}\sin{4x}\right)' + \left(\frac{3}{4}x\right)' + 0 = \frac{1}{8}(\cos{4x} \cdot 4) + \frac{3}{4} = \frac{1}{2}\cos{4x} + \frac{3}{4}$.
Теперь преобразуем функцию $g(x) = \sin^4{x} + \cos^4{x}$:
$g(x) = (\sin^2{x} + \cos^2{x})^2 - 2\sin^2{x}\cos^2{x} = 1^2 - 2(\sin{x}\cos{x})^2 = 1 - 2\left(\frac{\sin{2x}}{2}\right)^2 = 1 - 2\frac{\sin^2{2x}}{4} = 1 - \frac{1}{2}\sin^2{2x}$.
Используем формулу понижения степени $\sin^2{\alpha} = \frac{1-\cos{2\alpha}}{2}$:
$g(x) = 1 - \frac{1}{2}\left(\frac{1-\cos{4x}}{2}\right) = 1 - \frac{1-\cos{4x}}{4} = \frac{4 - (1-\cos{4x})}{4} = \frac{3+\cos{4x}}{4} = \frac{3}{4} + \frac{1}{4}\cos{4x}$.
Приравняем $g(x)$ к $f'(x)$:
$\frac{3}{4} + \frac{1}{4}\cos{4x} = \frac{1}{2}\cos{4x} + \frac{3}{4}$
$\frac{1}{4}\cos{4x} = \frac{1}{2}\cos{4x}$
$\frac{1}{2}\cos{4x} - \frac{1}{4}\cos{4x} = 0$
$\frac{1}{4}\cos{4x} = 0$
$\cos{4x} = 0$
$4x = \frac{\pi}{2} + \pi k, \quad k \in Z$
$x = \frac{\pi}{8} + \frac{\pi k}{4}, \quad k \in Z$
Ответ: $x = \frac{\pi}{8} + \frac{\pi k}{4}, \quad k \in Z$.
Другие задания:
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