Quiz — 4. DA rivation 3 Corrections..pdf

Question 1 sur 10 20:00

[{"id":30850,"question":"Quelle est la dérivée de la fonction f(x) = 3x² - 5x + 2 ?","option_a":"6x - 5","option_b":"3x - 5","option_c":"6x² - 5","option_d":"6x - 5x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de 3x² est 6x, celle de -5x est -5, et celle de 2 est 0. Ainsi, f'(x) = 6x - 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\": \"6x - 5\", \"b\": \"3x - 5\", \"c\": \"6x² - 5\", \"d\": \"6x - 5x + 2\"}}","_debug_options_count":4},{"id":30851,"question":"La dérivée de la fonction f(x) = |x| en x = 0 existe-t-elle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction valeur absolue n'est pas dérivable en x = 0 car elle présente un point anguleux à cet endroit (la dérivée à gauche et à droite ne sont pas égales).","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":30852,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"2e^(2x)","option_b":"e^(2x)","option_c":"2x*e^(2x)","option_d":"e^(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^(u(x)) est 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\": \"a\", \"options\": {\"a\": \"2e^(2x)\", \"b\": \"e^(2x)\", \"c\": \"2x*e^(2x)\", \"d\": \"e^(x)\"}}","_debug_options_count":4},{"id":30853,"question":"La dérivée d'une fonction constante est toujours égale à :","option_a":"1","option_b":"0","option_c":"x","option_d":"indéterminée","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction constante f(x) = k a une dérivée nulle car son taux d'accroissement est toujours égal à 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\": \"1\", \"b\": \"0\", \"c\": \"x\", \"d\": \"indéterminée\"}}","_debug_options_count":4},{"id":30854,"question":"La fonction f(x) = x^3 est-elle dérivable sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = x^3 est un polynôme, donc elle est dérivable sur tout ℝ. Sa dérivée est f'(x) = 3x².","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":30855,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x + 1) ?","option_a":"3\/(3x + 1)","option_b":"1\/(3x + 1)","option_c":"3x\/(3x + 1)","option_d":"1\/(x + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici, u(x) = 3x + 1, donc u'(x) = 3. Ainsi, f'(x) = 3\/(3x + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3\/(3x + 1)\", \"b\": \"1\/(3x + 1)\", \"c\": \"3x\/(3x + 1)\", \"d\": \"1\/(x + ","_debug_options_count":4},{"id":30856,"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":"Si f est paire (f(-x) = f(x)), alors f'(-x) = -f'(x), donc f' est 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},{"id":30857,"question":"Quelle est la dérivée de la fonction f(x) = sin(2x) ?","option_a":"cos(2x)","option_b":"2cos(2x)","option_c":"cos(x)","option_d":"2sin(2x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de sin(u(x)) est u'(x)*cos(u(x)). Ici, u(x) = 2x, donc u'(x) = 2. Ainsi, f'(x) = 2cos(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\": \"cos(2x)\", \"b\": \"2cos(2x)\", \"c\": \"cos(x)\", \"d\": \"2sin(2x)\"}}","_debug_options_count":4},{"id":30858,"question":"La dérivée d'une fonction strictement croissante est toujours positive.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si une fonction est strictement croissante sur un intervalle, alors sa dérivée est strictement positive sur cet intervalle (sauf éventuellement en des points isolés où elle pourrait être nulle).","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":30859,"question":"Quelle est la dérivée de la fonction f(x) = (2x + 1)\/(x - 3) ?","option_a":"2\/(x - 3)","option_b":"(2x - 7)\/(x - 3)²","option_c":"2\/(x - 3)²","option_d":"(x - 3)\/(2x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"On utilise la formule de dérivation d'un quotient : (u\/v)' = (u'v - uv')\/v². Ici, u = 2x + 1 (u' = 2) et v = x - 3 (v' = 1). Ainsi, f'(x) = (2(x - 3) - (2x + 1)*1)\/(x - 3)² = (2x - 6 - 2x - 1)\/(x - 3)² = -7\/(x - 3)².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2\/(x - 3)\", \"b\": \"(2x - 7)\/(x - 3)²\", \"c\": \"2\/(x - 3)²\", \"d\": \"","_debug_options_count":4}]