// Numbas version: finer_feedback_settings
{"name": "CINCO SEIS Derivada producto", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "s1", "b", "m", "n"], "name": "CINCO SEIS Derivada producto", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>Si &nbsp;$u$ y $v$ son funciones derivables de&nbsp;$x$,&nbsp;entonces la derivada del producto es:<br/>\\[\\simplify{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]</p>\n<p>En el ejercicio se tiene que:</p>\n<p>\\[\\simplify[dPoly]{u = ({a} + {b} * x) ^ {m}}\\Rightarrow \\simplify[dPoly]{Diff(u,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1}}\\]</p>\n<p>\\[\\simplify{v = e ^ ({n} * x)} \\Rightarrow \\simplify{Diff(v,x,1) = {n} * e ^ ({n} * x)}\\]</p>\n<p>Reemplazando en la f&oacute;rmula se obtiene:</p>\n<p>\\[\\simplify[dPoly]{Diff(f,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) + {n} * ({a} + {b} * x) ^ {m} * e ^ ({n} * x) = ({a} + {b} * x) ^ {m -1} * ({m * b + n * a} + {n * b} * x) * e ^ ({n} * x)}\\]</p>\n<p>En el &uacute;ltimo paso se ha factorizado la expresi&oacute;n, tomando factor com&uacute;n:&nbsp;$\\simplify[dPoly]{({a} + {b} * x) ^ {m -1} * e ^ ({n} * x)}$.</p>\n<p>De lo anterior se puede concluir que \\[\\simplify[dPoly]{g(x) = {m * b + n * a} + {n * b} * x}\\].</p>", "rulesets": {"std": ["all", "!collectNumbers"], "dpoly": ["std", "fractionNumbers"]}, "parts": [{"stepsPenalty": 0, "prompt": "<p>La derivada de la funci&oacute;n $\\simplify[dPoly]{f(x) = ({a} + {b} * x) ^ {m} * e ^ ({n} * x)}$ es: &nbsp;\\[\\simplify[dPoly]{Diff(f,x,1) = ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) * g(x)}\\]</p>\n<p>Determine el polinomio $g(x)$ faltante en la expresi&oacute;n.</p>\n<p>$g(x)=\\;$[[0]]</p>\n<code>Ejemplo de respuesta: escriba 7-8x para ingresar una respuesta como&nbsp;</code>$7-8x$", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"vsetrangepoints": 5, "prompt": "<p>Si &nbsp;$u$ y $v$ son funciones derivables de&nbsp;$x$,&nbsp;entonces la derivada del producto es:<br/>\\[\\simplify{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "", "marks": 0, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "dPoly", "scripts": {}, "answer": "({((m * b) + (n * a))} + ({(n * b)} * x))", "marks": "50", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "statement": "<p>Calcular la derivada empleando la regla del Producto.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "b": {"definition": "s1*random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "m": {"definition": "random(2..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "n": {"definition": "random(2..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}}, "metadata": {"description": "<p>Differentiate the function $f(x)=(a + b x)^m &nbsp;e ^ {n x}$&nbsp;using the product rule. Find $g(x)$ such that $f^{\\prime}(x)= (a + b x)^{m-1} &nbsp;e ^ {n x}g(x)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "contributors": [{"name": "Marlon Arcila", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/321/"}]}]}], "contributors": [{"name": "Marlon Arcila", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/321/"}]}