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[{"id":19858,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2x","option_c":"6x + 2x - 5","option_d":"6x² + 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'une fonction polynomiale se calcule terme par terme : (3x²)' = 6x, (2x)' = 2, et (-5)' = 0. Donc f'(x) = 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":19859,"question":"La dérivée de 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":"La dérivée de la fonction sinus est bien la fonction cosinus : (sin x)' = cos x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":19860,"question":"Quelle est la dérivée de f(x) = (2x + 1)\/(x² + 3) ?","option_a":"(2x² + 6x - 2)\/(x² + 3)²","option_b":"(2x² + 6x + 2)\/(x² + 3)²","option_c":"(2x² - 6x + 2)\/(x² + 3)²","option_d":"(2x² + 6x - 2)\/(x + 3)²","option_e":"","option_f":"","bonne_reponse":"A","explication":"On utilise la règle du quotient : (u\/v)' = (u'v - uv')\/v². Ici u = 2x + 1, u' = 2, v = x² + 3, v' = 2x. Le résultat est (2(x² + 3) - (2x + 1)(2x))\/(x² + 3)² = (2x² + 6 - 4x² - 2x)\/(x² + 3)² = (-2x² - 2x + 6)\/(x² + 3)².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":19861,"question":"La dérivée de f(x) = e^(3x) est f'(x) = 3e^(3x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'une fonction exponentielle composée est (e^u)' = u'e^u. Ici u = 3x, donc u' = 3. Ainsi f'(x) = 3e^(3x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":19862,"question":"Quelle est la dérivée de f(x) = ln(5x + 2) ?","option_a":"5\/(5x + 2)","option_b":"1\/(5x + 2)","option_c":"5x\/(5x + 2)","option_d":"1\/(x + 2)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de ln(u) est u'\/u. Ici u = 5x + 2, donc u' = 5. Ainsi f'(x) = 5\/(5x + 2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":19863,"question":"La dérivée de f(x) = x^4 * sin(x) est f'(x) = 4x^3 * sin(x) + x^4 * cos(x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"On utilise la règle du produit : (uv)' = u'v + uv'. Ici u = x^4, u' = 4x^3, v = sin(x), v' = cos(x). Donc f'(x) = 4x^3 * sin(x) + x^4 * cos(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":19864,"question":"Quelle est la dérivée de f(x) = (x² + 1)^3 ?","option_a":"3(x² + 1)^2","option_b":"6x(x² + 1)^2","option_c":"3x(x² + 1)^2","option_d":"6(x² + 1)^2","option_e":"","option_f":"","bonne_reponse":"B","explication":"On utilise la règle de la chaîne : (u^n)' = n*u^(n-1)*u'. Ici u = x² + 1, n = 3, u' = 2x. Donc f'(x) = 3*(x² + 1)^2 * 2x = 6x(x² + 1)^2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":19865,"question":"La dérivée de f(x) = tan(x) est f'(x) = 1 + tan²(x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de la fonction tangente est (tan x)' = 1 + tan²(x). Cette identité est souvent utilisée en trigonométrie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":19866,"question":"Quelle est la dérivée de f(x) = √(4x + 1) ?","option_a":"2\/√(4x + 1)","option_b":"1\/√(4x + 1)","option_c":"4\/√(4x + 1)","option_d":"1\/(2√(4x + 1))","option_e":"","option_f":"","bonne_reponse":"D","explication":"On utilise la règle de la chaîne pour les racines carrées : (√u)' = u'\/(2√u). Ici u = 4x + 1, u' = 4. Donc f'(x) = 4\/(2√(4x + 1)) = 2\/√(4x + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":19867,"question":"La dérivée de f(x) = cos(2x + π) est f'(x) = -2sin(2x + π).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de cos(u) est -u'sin(u). Ici u = 2x + π, donc u' = 2. Ainsi f'(x) = -2sin(2x + π).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]