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Номер 16.8, страница 96 - гдз по алгебре 10 класс учебник Абылкасымова, Жумагулова
Авторы: Абылкасымова А. Е., Жумагулова З. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
ISBN: 978-601-07-1142-6
Утверждено Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Глава 5. Производная. Параграф 16. Производная сложной функции - номер 16.8, страница 96.
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" => 1113873 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/16-8" "field_display_title" => "16.8" "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 {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #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" => 1113571 "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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #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 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "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 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "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" => "77" "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 {#1078 #items: array:1 [ 0 => App\Models\Book {#1049 #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" => 4293 "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 {#1051 #items: array:1 [ 0 => App\Models\Term {#1048 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ 0 => App\Models\Term {#1053 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ 0 => App\Models\Term {#1055 #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" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:2 [ 0 => App\Models\Term {#1057 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1058 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060 #items: array:1 [ 0 => App\Models\Term {#1061 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1062 #items: array:1 [ 0 => App\Models\Term {#1063 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#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:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4293 "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 {#1051} "field_class" => Illuminate\Database\Eloquent\Collection {#1052} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054} "field_author" => Illuminate\Database\Eloquent\Collection {#1056} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1059} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060} "field_country" => Illuminate\Database\Eloquent\Collection {#1062} "field_city" => Illuminate\Database\Eloquent\Collection {#1064} "field_series" => Illuminate\Database\Eloquent\Collection {#1066} "field_umk" => Illuminate\Database\Eloquent\Collection {#1067} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074} "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113571 "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 {#1050} "field_page_start" => "77" "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 {#1078} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:2 [ 0 => App\Models\Branch {#1113 #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" => 1113571 "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 {#1114 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "77" "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 {#1115 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1116 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113571 "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 {#1114} "field_page_start" => "77" "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 {#1115} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1116} ] #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 {#1117 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113576 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Производная сложной функции" "field_branch_order" => "16" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Term {#1118 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "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 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "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" => "94" "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 {#1120 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113576 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Производная сложной функции" "field_branch_order" => "16" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121} "field_page_start" => "94" "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 {#1120} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1119} ] #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 {#1102 #items: array:3 [ 0 => App\Models\Element {#1095 #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" => 1290842 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Edition {#1094 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1092 #items: array:1 [ 0 => App\Models\Term {#1093 #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_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1092} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1090} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1113873" "type" => "task" ] "text" => "<p><strong>16.8. a)</strong> $f(x) = \frac{12}{x-\sqrt{x}} - (x-6)^2$;</p><p><strong>б)</strong> $f(x) = \frac{10}{x+\sqrt{x}} + (\sqrt{x}-1)^3$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "2477" "height" => 313 "path" => "/media/algebra_10/Abylkasymova-u/0-16/8-1.webp?ts=1753263099" ] ] ] #original: array:7 [ "id" => 1290842 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087} "task" => array:2 [ "refs" => "1113873" "type" => "task" ] "text" => "<p><strong>16.8. a)</strong> $f(x) = \frac{12}{x-\sqrt{x}} - (x-6)^2$;</p><p><strong>б)</strong> $f(x) = \frac{10}{x+\sqrt{x}} + (\sqrt{x}-1)^3$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "2477" "height" => 313 "path" => "/media/algebra_10/Abylkasymova-u/0-16/8-1.webp?ts=1753263099" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1089 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1291356 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Edition {#1086 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Term {#1085 #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_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1084} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1083} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1113873" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1888" "height" => 1718 "path" => "/media/algebra_10/Abylkasymova-u/1-16/8-1.webp?ts=1753263296" ] ] ] #original: array:6 [ "id" => 1291356 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079} "task" => array:2 [ "refs" => "1113873" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1888" "height" => 1718 "path" => "/media/algebra_10/Abylkasymova-u/1-16/8-1.webp?ts=1753263296" ] ] ] #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\Element {#1081 #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" => 1549485 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1077 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1041 #items: array:1 [ 0 => App\Models\Term {#1043 #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: [] …3 } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1034 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1041} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1034} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1039} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1113873" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Рассмотрим функцию $f(x) = \frac{12}{x - \sqrt{x}} - (x - 6)^2$.</p><p>Область определения функции находится из условий: $x \ge 0$ (из-за $\sqrt{x}$) и $x - \sqrt{x} \neq 0$. Решая второе условие, получаем $\sqrt{x}(\sqrt{x}-1) \neq 0$, что дает $x \neq 0$ и $x \neq 1$. Таким образом, область определения: $x \in (0, 1) \cup (1, +\infty)$.</p><p>Для нахождения точек экстремума необходимо найти производную функции и приравнять ее к нулю. Для упрощения дифференцирования преобразуем первое слагаемое, разложив его на простейшие дроби. Введем замену $t = \sqrt{x}$.</p><p>$\frac{12}{x - \sqrt{x}} = \frac{12}{t^2 - t} = \frac{12}{t(t-1)}$.</p><p>Представим дробь в виде суммы: $\frac{12}{t(t-1)} = \frac{A}{t-1} + \frac{B}{t}$.</p><p>Для нахождения коэффициентов $A$ и $B$ приведем к общему знаменателю: $12 = At + B(t-1)$.</p><p>Подставив $t=1$, получим $12 = A \cdot 1 \Rightarrow A = 12$.</p><p>Подставив $t=0$, получим $12 = B \cdot (-1) \Rightarrow B = -12$.</p><p>Таким образом, исходная функция имеет вид: $f(x) = \frac{12}{\sqrt{x}-1} - \frac{12}{\sqrt{x}} - (x-6)^2$.</p><p>Теперь найдем производную $f'(x)$:</p><p>$f'(x) = \left(\frac{12}{\sqrt{x}-1}\right)' - \left(\frac{12}{\sqrt{x}}\right)' - \left((x-6)^2\right)'$</p><p>$f'(x) = 12 \cdot (-1)(\sqrt{x}-1)^{-2} \cdot \frac{1}{2\sqrt{x}} - 12 \cdot \left(-\frac{1}{2}\right)x^{-3/2} - 2(x-6)$</p><p>$f'(x) = -\frac{6}{\sqrt{x}(\sqrt{x}-1)^2} + \frac{6}{x\sqrt{x}} - 2(x-6)$</p><p>Приведем первые два слагаемых к общему знаменателю $x\sqrt{x}(\sqrt{x}-1)^2$:</p><p>$f'(x) = \frac{-6x + 6(\sqrt{x}-1)^2}{x\sqrt{x}(\sqrt{x}-1)^2} - 2(x-6) = \frac{-6x + 6(x-2\sqrt{x}+1)}{x\sqrt{x}(\sqrt{x}-1)^2} - 2(x-6)$</p><p>$f'(x) = \frac{-6x + 6x - 12\sqrt{x} + 6}{x\sqrt{x}(\sqrt{x}-1)^2} - 2(x-6) = \frac{6(1-2\sqrt{x})}{x\sqrt{x}(\sqrt{x}-1)^2} - 2(x-6)$</p><p>Приравняем производную к нулю для нахождения критических точек: $f'(x) = 0$.</p><p>$\frac{6(1-2\sqrt{x})}{x\sqrt{x}(\sqrt{x}-1)^2} - 2(x-6) = 0$</p><p>$\frac{3(1-2\sqrt{x})}{x\sqrt{x}(\sqrt{x}-1)^2} = x-6$</p><p>Данное трансцендентное уравнение не имеет простых аналитических решений. Его корни, которые и являются точками экстремума, могут быть найдены лишь численными методами.</p><p><strong>Ответ:</strong> Точки экстремума функции являются решениями уравнения $\frac{3(1 - 2\sqrt{x})}{x\sqrt{x}(\sqrt{x}-1)^2} = x-6$.</p><p><strong>б)</strong></p><p>Рассмотрим функцию $f(x) = \frac{10}{x + \sqrt{x}} + (\sqrt{x} - 1)^3$.</p><p>Область определения функции: $x + \sqrt{x} \neq 0$ и $x \ge 0$. Условие $x+\sqrt{x} = \sqrt{x}(\sqrt{x}+1) \neq 0$ при $x \ge 0$ эквивалентно $x \neq 0$. Таким образом, область определения: $x \in (0, +\infty)$.</p><p>Для упрощения найдем производную, используя замену $t = \sqrt{x}$, где $t > 0$. Тогда $x=t^2$ и $f(x) = F(t) = \frac{10}{t^2 + t} + (t-1)^3$.</p><p>Разложим первое слагаемое на простейшие дроби: $\frac{10}{t(t+1)} = \frac{A}{t} + \frac{B}{t+1}$.</p><p>Приводя к общему знаменателю, получаем $10 = A(t+1) + Bt$.</p><p>При $t=0$, $A=10$. При $t=-1$, $B=-10$.</p><p>Итак, $F(t) = \frac{10}{t} - \frac{10}{t+1} + (t-1)^3$.</p><p>Найдем производную $f'(x)$ по правилу производной сложной функции: $f'(x) = \frac{dF}{dt} \cdot \frac{dt}{dx}$.</p><p>$\frac{dt}{dx} = \frac{1}{2\sqrt{x}} = \frac{1}{2t}$.</p><p>$\frac{dF}{dt} = \left(\frac{10}{t}\right)' - \left(\frac{10}{t+1}\right)' + \left((t-1)^3\right)' = -\frac{10}{t^2} + \frac{10}{(t+1)^2} + 3(t-1)^2$.</p><p>Точки экстремума функции $f(x)$ соответствуют $f'(x) = 0$. Поскольку $\frac{dt}{dx} = \frac{1}{2t} > 0$ на области определения, то $f'(x)=0$ тогда и только тогда, когда $\frac{dF}{dt}=0$.</p><p>$-\frac{10}{t^2} + \frac{10}{(t+1)^2} + 3(t-1)^2 = 0$</p><p>$10\left(\frac{1}{(t+1)^2} - \frac{1}{t^2}\right) + 3(t-1)^2 = 0$</p><p>$10\frac{t^2 - (t+1)^2}{t^2(t+1)^2} + 3(t-1)^2 = 0$</p><p>$10\frac{t^2 - (t^2+2t+1)}{t^2(t+1)^2} + 3(t-1)^2 = 0$</p><p>$\frac{-10(2t+1)}{t^2(t+1)^2} + 3(t-1)^2 = 0$</p><p>$3(t-1)^2 = \frac{10(2t+1)}{t^2(t+1)^2}$</p><p>$3t^2(t-1)^2(t+1)^2 = 10(2t+1)$</p><p>$3(t(t-1)(t+1))^2 = 20t+10$</p><p>$3(t(t^2-1))^2 = 20t+10 \Rightarrow 3(t^3-t)^2 = 20t+10$</p><p>$3(t^6-2t^4+t^2) = 20t+10$</p><p>$3t^6-6t^4+3t^2-20t-10=0$</p><p>Получено полиномиальное уравнение шестой степени относительно $t$. Это уравнение не решается аналитически в общем виде. Его положительные корни могут быть найдены численными методами.</p><p><strong>Ответ:</strong> Точки экстремума функции $x=t^2$ определяются положительными корнями $t$ уравнения $3t^6-6t^4+3t^2-20t-10=0$.</p>" ] #original: array:6 [ "id" => 1549485 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1113873" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Рассмотрим функцию $f(x) = \frac{12}{x - \sqrt{x}} - (x - 6)^2$.</p><p>Область определения функции находится из условий: $x \ge 0$ (из-за $\sqrt{x}$) и $x - \sqrt{x} \neq 0$. Решая второе условие, получаем $\sqrt{x}(\sqrt{x}-1) \neq 0$, что дает $x \neq 0$ и $x \neq 1$. Таким образом, область определения: $x \in (0, 1) \cup (1, +\infty)$.</p><p>Для нахождения точек экстремума необходимо найти производную функции и приравнять ее к нулю. Для упрощения дифференцирования преобразуем первое слагаемое, разложив его на простейшие дроби. Введем замену $t = \sqrt{x}$.</p><p>$\frac{12}{x - \sqrt{x}} = \frac{12}{t^2 - t} = \frac{12}{t(t-1)}$.</p><p>Представим дробь в виде суммы: $\frac{12}{t(t-1)} = \frac{A}{t-1} + \frac{B}{t}$.</p><p>Для нахождения коэффициентов $A$ и $B$ приведем к общему знаменателю: $12 = At + B(t-1)$.</p><p>Подставив $t=1$, получим $12 = A \cdot 1 \Rightarrow A = 12$.</p><p>Подставив $t=0$, получим $12 = B \cdot (-1) \Rightarrow B = -12$.</p><p>Таким образом, исходная функция имеет вид: $f(x) = \frac{12}{\sqrt{x}-1} - \frac{12}{\sqrt{x}} - (x-6)^2$.</p><p>Теперь найдем производную $f'(x)$:</p><p>$f'(x) = \left(\frac{12}{\sqrt{x}-1}\right)' - \left(\frac{12}{\sqrt{x}}\right)' - \left((x-6)^2\right)'$</p><p>$f'(x) = 12 \cdot (-1)(\sqrt{x}-1)^{-2} \cdot \frac{1}{2\sqrt{x}} - 12 \cdot \left(-\frac{1}{2}\right)x^{-3/2} - 2(x-6)$</p><p>$f'(x) = -\frac{6}{\sqrt{x}(\sqrt{x}-1)^2} + \frac{6}{x\sqrt{x}} - 2(x-6)$</p><p>Приведем первые два слагаемых к общему знаменателю $x\sqrt{x}(\sqrt{x}-1)^2$:</p><p>$f'(x) = \frac{-6x + 6(\sqrt{x}-1)^2}{x\sqrt{x}(\sqrt{x}-1)^2} - 2(x-6) = \frac{-6x + 6(x-2\sqrt{x}+1)}{x\sqrt{x}(\sqrt{x}-1)^2} - 2(x-6)$</p><p>$f'(x) = \frac{-6x + 6x - 12\sqrt{x} + 6}{x\sqrt{x}(\sqrt{x}-1)^2} - 2(x-6) = \frac{6(1-2\sqrt{x})}{x\sqrt{x}(\sqrt{x}-1)^2} - 2(x-6)$</p><p>Приравняем производную к нулю для нахождения критических точек: $f'(x) = 0$.</p><p>$\frac{6(1-2\sqrt{x})}{x\sqrt{x}(\sqrt{x}-1)^2} - 2(x-6) = 0$</p><p>$\frac{3(1-2\sqrt{x})}{x\sqrt{x}(\sqrt{x}-1)^2} = x-6$</p><p>Данное трансцендентное уравнение не имеет простых аналитических решений. Его корни, которые и являются точками экстремума, могут быть найдены лишь численными методами.</p><p><strong>Ответ:</strong> Точки экстремума функции являются решениями уравнения $\frac{3(1 - 2\sqrt{x})}{x\sqrt{x}(\sqrt{x}-1)^2} = x-6$.</p><p><strong>б)</strong></p><p>Рассмотрим функцию $f(x) = \frac{10}{x + \sqrt{x}} + (\sqrt{x} - 1)^3$.</p><p>Область определения функции: $x + \sqrt{x} \neq 0$ и $x \ge 0$. Условие $x+\sqrt{x} = \sqrt{x}(\sqrt{x}+1) \neq 0$ при $x \ge 0$ эквивалентно $x \neq 0$. Таким образом, область определения: $x \in (0, +\infty)$.</p><p>Для упрощения найдем производную, используя замену $t = \sqrt{x}$, где $t > 0$. Тогда $x=t^2$ и $f(x) = F(t) = \frac{10}{t^2 + t} + (t-1)^3$.</p><p>Разложим первое слагаемое на простейшие дроби: $\frac{10}{t(t+1)} = \frac{A}{t} + \frac{B}{t+1}$.</p><p>Приводя к общему знаменателю, получаем $10 = A(t+1) + Bt$.</p><p>При $t=0$, $A=10$. При $t=-1$, $B=-10$.</p><p>Итак, $F(t) = \frac{10}{t} - \frac{10}{t+1} + (t-1)^3$.</p><p>Найдем производную $f'(x)$ по правилу производной сложной функции: $f'(x) = \frac{dF}{dt} \cdot \frac{dt}{dx}$.</p><p>$\frac{dt}{dx} = \frac{1}{2\sqrt{x}} = \frac{1}{2t}$.</p><p>$\frac{dF}{dt} = \left(\frac{10}{t}\right)' - \left(\frac{10}{t+1}\right)' + \left((t-1)^3\right)' = -\frac{10}{t^2} + \frac{10}{(t+1)^2} + 3(t-1)^2$.</p><p>Точки экстремума функции $f(x)$ соответствуют $f'(x) = 0$. Поскольку $\frac{dt}{dx} = \frac{1}{2t} > 0$ на области определения, то $f'(x)=0$ тогда и только тогда, когда $\frac{dF}{dt}=0$.</p><p>$-\frac{10}{t^2} + \frac{10}{(t+1)^2} + 3(t-1)^2 = 0$</p><p>$10\left(\frac{1}{(t+1)^2} - \frac{1}{t^2}\right) + 3(t-1)^2 = 0$</p><p>$10\frac{t^2 - (t+1)^2}{t^2(t+1)^2} + 3(t-1)^2 = 0$</p><p>$10\frac{t^2 - (t^2+2t+1)}{t^2(t+1)^2} + 3(t-1)^2 = 0$</p><p>$\frac{-10(2t+1)}{t^2(t+1)^2} + 3(t-1)^2 = 0$</p><p>$3(t-1)^2 = \frac{10(2t+1)}{t^2(t+1)^2}$</p><p>$3t^2(t-1)^2(t+1)^2 = 10(2t+1)$</p><p>$3(t(t-1)(t+1))^2 = 20t+10$</p><p>$3(t(t^2-1))^2 = 20t+10 \Rightarrow 3(t^3-t)^2 = 20t+10$</p><p>$3(t^6-2t^4+t^2) = 20t+10$</p><p>$3t^6-6t^4+3t^2-20t-10=0$</p><p>Получено полиномиальное уравнение шестой степени относительно $t$. Это уравнение не решается аналитически в общем виде. Его положительные корни могут быть найдены численными методами.</p><p><strong>Ответ:</strong> Точки экстремума функции $x=t^2$ определяются положительными корнями $t$ уравнения $3t^6-6t^4+3t^2-20t-10=0$.</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" => "1113874" "type" => "task" ] "previous" => array:2 [ "refs" => "1113872" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\BookPage {#1042 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: 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№16.8 (с. 96)
Условие. №16.8 (с. 96)
скриншот условия
16.8. a) $f(x) = \frac{12}{x-\sqrt{x}} - (x-6)^2$;
б) $f(x) = \frac{10}{x+\sqrt{x}} + (\sqrt{x}-1)^3$.
Решение. №16.8 (с. 96)
Решение 2. №16.8 (с. 96)
а)
Рассмотрим функцию $f(x) = \frac{12}{x - \sqrt{x}} - (x - 6)^2$.
Область определения функции находится из условий: $x \ge 0$ (из-за $\sqrt{x}$) и $x - \sqrt{x} \neq 0$. Решая второе условие, получаем $\sqrt{x}(\sqrt{x}-1) \neq 0$, что дает $x \neq 0$ и $x \neq 1$. Таким образом, область определения: $x \in (0, 1) \cup (1, +\infty)$.
Для нахождения точек экстремума необходимо найти производную функции и приравнять ее к нулю. Для упрощения дифференцирования преобразуем первое слагаемое, разложив его на простейшие дроби. Введем замену $t = \sqrt{x}$.
$\frac{12}{x - \sqrt{x}} = \frac{12}{t^2 - t} = \frac{12}{t(t-1)}$.
Представим дробь в виде суммы: $\frac{12}{t(t-1)} = \frac{A}{t-1} + \frac{B}{t}$.
Для нахождения коэффициентов $A$ и $B$ приведем к общему знаменателю: $12 = At + B(t-1)$.
Подставив $t=1$, получим $12 = A \cdot 1 \Rightarrow A = 12$.
Подставив $t=0$, получим $12 = B \cdot (-1) \Rightarrow B = -12$.
Таким образом, исходная функция имеет вид: $f(x) = \frac{12}{\sqrt{x}-1} - \frac{12}{\sqrt{x}} - (x-6)^2$.
Теперь найдем производную $f'(x)$:
$f'(x) = \left(\frac{12}{\sqrt{x}-1}\right)' - \left(\frac{12}{\sqrt{x}}\right)' - \left((x-6)^2\right)'$
$f'(x) = 12 \cdot (-1)(\sqrt{x}-1)^{-2} \cdot \frac{1}{2\sqrt{x}} - 12 \cdot \left(-\frac{1}{2}\right)x^{-3/2} - 2(x-6)$
$f'(x) = -\frac{6}{\sqrt{x}(\sqrt{x}-1)^2} + \frac{6}{x\sqrt{x}} - 2(x-6)$
Приведем первые два слагаемых к общему знаменателю $x\sqrt{x}(\sqrt{x}-1)^2$:
$f'(x) = \frac{-6x + 6(\sqrt{x}-1)^2}{x\sqrt{x}(\sqrt{x}-1)^2} - 2(x-6) = \frac{-6x + 6(x-2\sqrt{x}+1)}{x\sqrt{x}(\sqrt{x}-1)^2} - 2(x-6)$
$f'(x) = \frac{-6x + 6x - 12\sqrt{x} + 6}{x\sqrt{x}(\sqrt{x}-1)^2} - 2(x-6) = \frac{6(1-2\sqrt{x})}{x\sqrt{x}(\sqrt{x}-1)^2} - 2(x-6)$
Приравняем производную к нулю для нахождения критических точек: $f'(x) = 0$.
$\frac{6(1-2\sqrt{x})}{x\sqrt{x}(\sqrt{x}-1)^2} - 2(x-6) = 0$
$\frac{3(1-2\sqrt{x})}{x\sqrt{x}(\sqrt{x}-1)^2} = x-6$
Данное трансцендентное уравнение не имеет простых аналитических решений. Его корни, которые и являются точками экстремума, могут быть найдены лишь численными методами.
Ответ: Точки экстремума функции являются решениями уравнения $\frac{3(1 - 2\sqrt{x})}{x\sqrt{x}(\sqrt{x}-1)^2} = x-6$.
б)
Рассмотрим функцию $f(x) = \frac{10}{x + \sqrt{x}} + (\sqrt{x} - 1)^3$.
Область определения функции: $x + \sqrt{x} \neq 0$ и $x \ge 0$. Условие $x+\sqrt{x} = \sqrt{x}(\sqrt{x}+1) \neq 0$ при $x \ge 0$ эквивалентно $x \neq 0$. Таким образом, область определения: $x \in (0, +\infty)$.
Для упрощения найдем производную, используя замену $t = \sqrt{x}$, где $t > 0$. Тогда $x=t^2$ и $f(x) = F(t) = \frac{10}{t^2 + t} + (t-1)^3$.
Разложим первое слагаемое на простейшие дроби: $\frac{10}{t(t+1)} = \frac{A}{t} + \frac{B}{t+1}$.
Приводя к общему знаменателю, получаем $10 = A(t+1) + Bt$.
При $t=0$, $A=10$. При $t=-1$, $B=-10$.
Итак, $F(t) = \frac{10}{t} - \frac{10}{t+1} + (t-1)^3$.
Найдем производную $f'(x)$ по правилу производной сложной функции: $f'(x) = \frac{dF}{dt} \cdot \frac{dt}{dx}$.
$\frac{dt}{dx} = \frac{1}{2\sqrt{x}} = \frac{1}{2t}$.
$\frac{dF}{dt} = \left(\frac{10}{t}\right)' - \left(\frac{10}{t+1}\right)' + \left((t-1)^3\right)' = -\frac{10}{t^2} + \frac{10}{(t+1)^2} + 3(t-1)^2$.
Точки экстремума функции $f(x)$ соответствуют $f'(x) = 0$. Поскольку $\frac{dt}{dx} = \frac{1}{2t} > 0$ на области определения, то $f'(x)=0$ тогда и только тогда, когда $\frac{dF}{dt}=0$.
$-\frac{10}{t^2} + \frac{10}{(t+1)^2} + 3(t-1)^2 = 0$
$10\left(\frac{1}{(t+1)^2} - \frac{1}{t^2}\right) + 3(t-1)^2 = 0$
$10\frac{t^2 - (t+1)^2}{t^2(t+1)^2} + 3(t-1)^2 = 0$
$10\frac{t^2 - (t^2+2t+1)}{t^2(t+1)^2} + 3(t-1)^2 = 0$
$\frac{-10(2t+1)}{t^2(t+1)^2} + 3(t-1)^2 = 0$
$3(t-1)^2 = \frac{10(2t+1)}{t^2(t+1)^2}$
$3t^2(t-1)^2(t+1)^2 = 10(2t+1)$
$3(t(t-1)(t+1))^2 = 20t+10$
$3(t(t^2-1))^2 = 20t+10 \Rightarrow 3(t^3-t)^2 = 20t+10$
$3(t^6-2t^4+t^2) = 20t+10$
$3t^6-6t^4+3t^2-20t-10=0$
Получено полиномиальное уравнение шестой степени относительно $t$. Это уравнение не решается аналитически в общем виде. Его положительные корни могут быть найдены численными методами.
Ответ: Точки экстремума функции $x=t^2$ определяются положительными корнями $t$ уравнения $3t^6-6t^4+3t^2-20t-10=0$.
Другие задания:
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