Testez vos connaissances sur la dérivabilité et les fonctions dérivées

Question 1 sur 10 10:00

[{"id":29318,"question":"Soit f(x) = x² + 3x - 5. Quelle est la dérivée f'(x) ?","option_a":"A. 2x + 3","option_b":"B. x² + 3","option_c":"C. 2x - 5","option_d":"D. 3x + 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de x² est 2x et celle de 3x est 3. La dérivée d'une constante est 0. Donc f'(x) = 2x + 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29319,"question":"La fonction f(x) = |x| est-elle dérivable en x = 0 ?","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 0 car elle n'admet pas de dérivée à gauche et à droite égales en ce point (dérivée à gauche = -1, dérivée à droite = 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29320,"question":"Quelle est la dérivée de f(x) = e^(2x) ?","option_a":"A. e^(2x)","option_b":"B. 2e^(2x)","option_c":"C. e^x","option_d":"D. 2e^x","option_e":"","option_f":"","bonne_reponse":"B","explication":"En utilisant la règle de dérivation des fonctions exponentielles, f'(x) = 2 * e^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29321,"question":"Soit f(x) = ln(x² + 1). Quelle est la dérivée f'(x) ?","option_a":"A. 2x \/ (x² + 1)","option_b":"B. 2x","option_c":"C. 1 \/ (x² + 1)","option_d":"D. x \/ (x² + 1)","option_e":"","option_f":"","bonne_reponse":"A","explication":"On utilise la règle de la chaîne : f'(x) = (1 \/ (x² + 1)) * 2x = 2x \/ (x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29322,"question":"La dérivée d'une fonction constante est toujours nulle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Oui, car la pente de la tangente à une fonction constante est nulle en tout point.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29323,"question":"Quelle est la dérivée de f(x) = sin(3x) ?","option_a":"A. cos(3x)","option_b":"B. 3cos(3x)","option_c":"C. -3cos(3x)","option_d":"D. sin(3x)","option_e":"","option_f":"","bonne_reponse":"B","explication":"En utilisant la règle de la chaîne, f'(x) = cos(3x) * 3 = 3cos(3x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29324,"question":"Soit f(x) = (x² + 1) \/ (x - 2). Quelle est la dérivée f'(x) ?","option_a":"A. (2x(x - 2) - (x² + 1)) \/ (x - 2)²","option_b":"B. (2x(x - 2) + (x² + 1)) \/ (x - 2)²","option_c":"C. (2x - (x² + 1)) \/ (x - 2)²","option_d":"D. (2x(x - 2) - (x² + 1)) \/ (x - 2)","option_e":"","option_f":"","bonne_reponse":"A","explication":"On utilise la règle du quotient : f'(x) = [2x(x - 2) - (x² + 1)] \/ (x - 2)².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29325,"question":"La dérivée d'une fonction paire est toujours impaire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Oui, car si f(-x) = f(x), alors f'(-x) = -f'(x), ce qui est la définition d'une fonction impaire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29326,"question":"Quelle est la dérivée de f(x) = x^3 * ln(x) ?","option_a":"A. 3x² * ln(x)","option_b":"B. x²(3ln(x) + 1)","option_c":"C. 3x² + ln(x)","option_d":"D. x²(3ln(x) - 1)","option_e":"","option_f":"","bonne_reponse":"B","explication":"On utilise la règle du produit : f'(x) = 3x² * ln(x) + x^3 * (1\/x) = x²(3ln(x) + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29327,"question":"Soit f(x) = √(x² + 4). Quelle est la dérivée f'(x) ?","option_a":"A. x \/ √(x² + 4)","option_b":"B. 2x \/ √(x² + 4)","option_c":"C. x \/ (2√(x² + 4))","option_d":"D. x \/ √(x² + 2)","option_e":"","option_f":"","bonne_reponse":"C","explication":"On utilise la règle de la chaîne : f'(x) = (1 \/ (2√(x² + 4))) * 2x = x \/ √(x² + 4).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]