// Numbas version: finer_feedback_settings
{"name": "Chain rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Chain rule", "tags": ["Calculus", "algebraic manipulation", "calculus", "chain rule", "derivative of the product of two functions", "differentiation", "product rule"], "advice": "\n    \n    \n    <p>The product rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify[std]{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[std]{u = x ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m}x ^ {m -1}}\\]</p>\n    \n    \n    \n    <p>\\[\\simplify[std]{v = ({a} * x^2+{b})^{n}} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {2*n*a}*x * ({a} * x^2+{b})^{n-1}}\\]</p>\n    \n    \n    \n    <p>For this last differentiation we used the chain rule.</p>\n    \n    \n    \n    <p>Hence on substituting into the product rule above we get:</p>\n    \n    \n    \n    <p>\\[\\begin{eqnarray*}\\frac{df}{dx} &#38;=&#38; \\simplify[std]{{m}x ^ {m-1} * ({a} * x^2+{b})^{n}+x^{m} *{2*n*a}*x* ({a} * x^2+{b})^{n-1}}\\\\\n    \n    &#38;=&#38; \\simplify[std]{{m}x ^ {m-1} * ({a} * x^2+{b})^{n}+{2*n*a}*x^{m+1}* ({a} * x^2+{b})^{n-1}}\\\\\n    \n    &#38;=&#38; \\simplify[std]{x ^ {m-1} * ({a} * x^2+{b})^{n-1}*({m}*({a}*x^2+{b})+{2*n*a}x^{2})} \\\\\n    \n    &#38;=&#38;\\simplify[std]{x ^ {m-1} * ({a} * x^2+{b})^{n-1}*({m*a+2*a*n}*x^2+{m*b})}\n    \n    \\end{eqnarray*}\\]</p>\n    \n    \n    \n    <p>Hence $\\simplify[std]{g(x)={m*a+2*a*n}*x^2+{m*b}}$</p>\n    \n    \n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n      <p>$\\simplify[std]{f(x) = x ^ {m} * ({a} * x^2+{b})^{n}}$<br/>The answer is in the form <br/>\\[\\frac{df}{dx}=\\simplify[std]{x^{m-1}({a}x^2+{b})^{n-1}*g(x)}\\] for a polynomial $g(x)$.</p>\n        <p>You have to find $g(x)$.</p>\n        <p>$g(x)=\\;$[[0]]</p>\n        <p>Click on Show steps for 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": "std", "marks": 3.0, "answer": "{m*a+2*a*n}*x^2+{m*b}", "type": "jme"}], "steps": [{"prompt": "<p>You should use the the product rule and the chain rule for this example.</p>", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "extensions": [], "statement": "<p>Differentiate the following function $f(x)$.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..9)", "name": "a"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b": {"definition": "s1*random(1..9)", "name": "b"}, "m": {"definition": "random(3..9)", "name": "m"}, "n": {"definition": "random(3..9)", "name": "n"}}, "metadata": {"notes": "\n                            \t\t<p><strong>1/08/2012:</strong></p>\n                            \t\t<p>Added tags.</p>\n                            \t\t<p>Added description.</p>\n                            \t\t<p>Checked calculation. OK.</p>\n                            \t\t<p>Added information about Show steps. Altered to 0 marks lost rather than 1.</p>\n                            \t\t<p>Got rid of a redundant ruleset.</p>\n                            \t\t<p>&nbsp;</p>\n                            \t\t<p>&nbsp;</p>\n                            \t\t", "description": "<p>The derivative of $\\displaystyle x ^ {m}(ax^2+b)^{n}$ is of the form $\\displaystyle x^{m-1}(ax^2+b)^{n-1}g(x)$. Find $g(x)$.</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/"}]}