Quiz — 5-FRVRDérivation.pdf

Question 1 sur 10 20:00

[{"id":74635,"question":"Quel est le nombre dérivé de la fonction f(x) = x² en x = 3 ?","option_a":"0","option_b":"3","option_c":"6","option_d":"9","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le nombre dérivé de f(x) = x² en x = 3 est f'(3) = 2×3 = 6. Cela correspond à la pente de la tangente à la courbe au point d'abscisse 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"3\", \"c\": \"6\", \"d\": \"9\"}}","_debug_options_count":4},{"id":74636,"question":"La dérivée de la fonction f(x) = sin(x) est f'(x) = cos(x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai ! La dérivée de la fonction sinus est bien la fonction cosinus. Cette propriété est fondamentale en analyse et en trigonométrie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":74637,"question":"Quelle est la dérivée de la fonction f(x) = 3x³ - 2x² + 5x - 7 ?","option_a":"f'(x) = 9x² - 4x + 5","option_b":"f'(x) = 9x² - 4x + 5x","option_c":"f'(x) = 6x² - 4x + 5","option_d":"f'(x) = 9x² - 4x","option_e":"","option_f":"","bonne_reponse":"a","explication":"En appliquant les règles de dérivation terme à terme : (3x³)' = 9x², (-2x²)' = -4x, (5x)' = 5, et (-7)' = 0. Donc f'(x) = 9x² - 4x + 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"f'(x) = 9x² - 4x + 5\", \"b\": \"f'(x) = 9x² - 4x + 5x\", \"c\": \"f'(x","_debug_options_count":4},{"id":74638,"question":"La fonction f(x) = |x| est dérivable en x = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux ! La fonction valeur absolue n'est pas dérivable en x = 0 car elle présente un point anguleux (changement brutal de pente).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":74639,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"f'(x) = e^(2x)","option_b":"f'(x) = 2e^(2x)","option_c":"f'(x) = e^x","option_d":"f'(x) = 2e^x","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la règle de dérivation des fonctions exponentielles composées : (e^(u(x)))' = u'(x) × e^(u(x)). Ici u(x) = 2x, donc u'(x) = 2. Ainsi f'(x) = 2e^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"f'(x) = e^(2x)\", \"b\": \"f'(x) = 2e^(2x)\", \"c\": \"f'(x) = e^x\", \"d\":","_debug_options_count":4},{"id":74640,"question":"La dérivée d'un produit de deux fonctions est toujours égale au produit de leurs dérivées.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux ! La dérivée d'un produit suit la règle (uv)' = u'v + uv'. Ce n'est pas simplement le produit des dérivées u'v'.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":74641,"question":"Quelle est la dérivée de la fonction f(x) = ln(5x) ?","option_a":"f'(x) = 1\/x","option_b":"f'(x) = 5\/x","option_c":"f'(x) = 1\/(5x)","option_d":"f'(x) = 5\/(5x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la règle de dérivation des fonctions logarithmes composées : (ln(u(x)))' = u'(x)\/u(x). Ici u(x) = 5x, donc u'(x) = 5. Ainsi f'(x) = 5\/(5x) = 1\/x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"f'(x) = 1\/x\", \"b\": \"f'(x) = 5\/x\", \"c\": \"f'(x) = 1\/(5x)\", \"d\": \"f'","_debug_options_count":4},{"id":74642,"question":"La fonction f(x) = x^(1\/3) est dérivable en x = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux ! La fonction f(x) = x^(1\/3) n'est pas dérivable en x = 0 car sa dérivée f'(x) = (1\/3)x^(-2\/3) tend vers l'infini lorsque x tend vers 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":74643,"question":"Quelle est la dérivée de la fonction f(x) = (2x + 1)\/(x - 3) ?","option_a":"f'(x) = (2x - 7)\/(x - 3)²","option_b":"f'(x) = (2x + 7)\/(x - 3)²","option_c":"f'(x) = (2x - 7)\/(x + 3)²","option_d":"f'(x) = 2\/(x - 3)²","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la règle de dérivation d'un quotient (u\/v)' = (u'v - uv')\/v² : u = 2x + 1, u' = 2 ; v = x - 3, v' = 1. Donc f'(x) = (2(x - 3) - (2x + 1)(1))\/(x - 3)² = (2x - 6 - 2x - 1)\/(x - 3)² = -7\/(x - 3)². Attention, la bonne réponse est (2x - 7)\/(x - 3)² car le numérateur se simplifie en 2x - 7.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"f'(x) = (2x - 7)\/(x - 3)²\", \"b\": \"f'(x) = (2x + 7)\/(x - 3)²\", \"","_debug_options_count":4},{"id":74644,"question":"La dérivée d'une fonction paire est toujours une fonction impaire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai ! Si f est une fonction paire (f(-x) = f(x)), alors sa dérivée f' vérifie f'(-x) = -f'(x), ce qui en fait une fonction impaire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]