// Numbas version: finer_feedback_settings
{"name": "Product Rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Product Rule", "tags": ["Steps", "algebraic manipulation", "derivative", "deriving a function", "differentiate", "differentiating a function", "differentiating a product of functions", "differentiation", "exponential function", "functions", "product rule", "steps"], "advice": "\n    \n    \n    <p>The product rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]</p>\n    \n    \n    \n    <p>For this example:</p>\n    \n    \n    \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    \n    \n    \n    <p>\\[\\simplify{v = e ^ ({n} * x)} \\Rightarrow \\simplify{Diff(v,x,1) = {n} * e ^ ({n} * x)}\\]</p>\n    \n    \n    \n    <p>Hence on substituting into the product rule above we get:</p>\n    \n    \n    \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    \n    \n    \n    <p>The last step was to take out the common term $\\simplify[dPoly]{({a} + {b} * x) ^ {m -1} * e ^ ({n} * x)}$.</p>\n    \n    \n    \n    <p>Hence \\[\\simplify[dPoly]{g(x) = {m * b + n * a} + {n * b} * x}\\].</p>\n    \n    \n    \n  ", "rulesets": {"std": ["all", "!collectNumbers"], "dpoly": ["std", "fractionNumbers"]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n      <p>$\\simplify[dPoly]{f(x) = ({a} + {b} * x) ^ {m} * e ^ ({n} * x)}$</p>\n        <p>You are given that \\[\\simplify[dPoly]{Diff(f,x,1) = ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) * g(x)}\\]</p>\n        <p>for a polynomial $g(x)$. You have to find $g(x)$.</p>\n        <p>$g(x)=\\;$[[0]]</p>\n        <p>Clicking on Show steps gives you more information, you will not lose any marks by doing so.</p>\n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "dPoly", "marks": 3.0, "answer": "({((m * b) + (n * a))} + ({(n * b)} * x))", "type": "jme"}], "steps": [{"prompt": "<p>The product rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]</p>", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "extensions": [], "statement": "<p>Differentiate the following function $f(x)$ using the product rule.</p>", "variable_groups": [], "progress": "testing", "type": "question", "variables": {"a": {"definition": "random(1..4)", "name": "a"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b": {"definition": "s1*random(1..5)", "name": "b"}, "m": {"definition": "random(2..8)", "name": "m"}, "n": {"definition": "random(2..6)", "name": "n"}}, "metadata": {"notes": "\n    \t\t<p><strong>20/06/2012:</strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p><strong>4/7/2012:<br /></strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p><strong>31/07/2012:</strong></p>\n    \t\t<p>Checked calculation.</p>\n    \t\t<p>Allowed no penalty on looking at Steps.</p>\n    \t\t", "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 $f\\;'(x)= (a + b x)^{m-1} &nbsp;e ^ {n x}g(x)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}