Quiz — سلسلة تمارين في الاشتقاق.pdf

Question 1 sur 10 20:00

[{"id":74170,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 5x - 7 ?","option_a":"6x + 5","option_b":"3x² + 5","option_c":"6x + 5x - 7","option_d":"3x + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme se calcule terme à terme : (3x²)' = 6x, (5x)' = 5, et (-7)' = 0. Donc 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\": \"3x² + 5\", \"c\": \"6x + 5x - 7\", \"d\": \"3x + 5\"}}","_debug_options_count":4},{"id":74171,"question":"La dérivée de la fonction f(x) = e^(2x) est f'(x) = 2e^(2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai : la dérivée de e^(u(x)) est u'(x)e^(u(x)), ici u(x)=2x donc u'(x)=2.","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":74172,"question":"Quelle est la dérivée de f(x) = (x² + 1)\/(x - 3) ?","option_a":"(2x(x-3) - (x²+1))\/(x-3)²","option_b":"(2x)\/(x-3)","option_c":"(x²+1)\/(x-3)²","option_d":"2x - 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise la formule de dérivation d'un quotient : (u\/v)' = (u'v - uv')\/v². Ici u=x²+1, v=x-3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"(2x(x-3) - (x²+1))\/(x-3)²\", \"b\": \"(2x)\/(x-3)\", \"c\": \"(x²+1)\/(x","_debug_options_count":4},{"id":74173,"question":"La dérivée de f(x) = ln(x² + 1) est f'(x) = 2x\/(x² + 1).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai : la dérivée de ln(u(x)) est u'(x)\/u(x), ici u(x)=x²+1 donc u'(x)=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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":74174,"question":"Quelle est la dérivée de f(x) = sin(3x + π\/4) ?","option_a":"3cos(3x + π\/4)","option_b":"cos(3x + π\/4)","option_c":"3sin(3x + π\/4)","option_d":"cos(x + π\/4)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Dérivée d'une fonction trigonométrique composée : (sin(u(x)))' = u'(x)cos(u(x)). Ici u(x)=3x+π\/4 donc u'(x)=3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3cos(3x + π\/4)\", \"b\": \"cos(3x + π\/4)\", \"c\": \"3sin(3x + π\/4)\", ","_debug_options_count":4},{"id":74175,"question":"La dérivée de f(x) = √(x² + 4) est f'(x) = x\/√(x² + 4).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai : la dérivée de √u(x) est u'(x)\/(2√u(x)). Ici u(x)=x²+4 donc u'(x)=2x, d'où f'(x)=2x\/(2√(x²+4))=x\/√(x²+4).","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":74176,"question":"Quelle est la dérivée de f(x) = x³e^x ?","option_a":"3x²e^x","option_b":"x²e^x(3 + x)","option_c":"3x² + e^x","option_d":"x²e^x(3 - x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"On utilise la règle du produit : (uv)' = u'v + uv'. Ici u=x³, v=e^x donc u'=3x², v'=e^x. Ainsi f'(x)=3x²e^x + x³e^x = x²e^x(3 + x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"3x²e^x\", \"b\": \"x²e^x(3 + x)\", \"c\": \"3x² + e^x\", \"d\": \"x²e^x(3","_debug_options_count":4},{"id":74177,"question":"La dérivée de f(x) = cos²(x) est f'(x) = -2cos(x)sin(x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai : on peut écrire f(x)=(cos(x))² et utiliser la règle de la chaîne : 2cos(x) × (-sin(x)) = -2cos(x)sin(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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":74178,"question":"Quelle est la dérivée de f(x) = (2x + 1)^5 ?","option_a":"5(2x+1)^4","option_b":"10(2x+1)^4","option_c":"2(2x+1)^4","option_d":"5(2x)^4","option_e":"","option_f":"","bonne_reponse":"b","explication":"Dérivée d'une fonction composée : 5(2x+1)^4 × 2 = 10(2x+1)^4 (règle de la chaîne).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"5(2x+1)^4\", \"b\": \"10(2x+1)^4\", \"c\": \"2(2x+1)^4\", \"d\": \"5(2x)^4\"}}","_debug_options_count":4},{"id":74179,"question":"La dérivée de f(x) = 1\/x est f'(x) = -1\/x².","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai : f(x)=x^(-1) donc f'(x)=-1x^(-2)=-1\/x².","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}]