// 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", "Steps", "calculus", "chain rule", "derivative of a function of a function", "differentiation", "function of a function", "steps"], "advice": "\n  \n  \n  <p>$\\simplify[std]{f(x) = sqrt({a} * x^{m}+{b})}$<br/>The chain rule says that if $f(x)=g(h(x))$ then<br/>\\[\\simplify[std]{f&#39;(x) = h&#39;(x)g&#39;(h(x))}\\]<br/>One way to find $f&#39;(x)$ is to let $u=h(x)$ then we have $f(u)=g(u)$ as a function of $u$.<br/>Then we use the chain rule in the form:<br/>\\[\\frac{df}{dx} = \\frac{du}{dx}\\frac{df(u)}{du}\\]<br/>Once you have worked this out, you replace $u$ by $h(x)$ and your answer is now in terms of $x$.</p>\n  \n  \n  \n  <p>For this example, we let $u=\\simplify[std]{{a} * x^{m}+{b}}$ and we have $f(u)=\\simplify[std]{sqrt(u)=u^{1/2}}$.<br/>This gives<br/>\\[\\begin{eqnarray*}\\frac{du}{dx} &#38;=&#38; \\simplify[std]{{m*a}x ^ {m -1}}\\\\\n  \n  \\frac{df(u)}{du} &#38;=&#38; \\simplify[std]{{1/2}*u^{-1/2}=1/(2*sqrt(u))} \\end{eqnarray*}\\]</p>\n  \n  \n  \n  <p>Hence on substituting into the chain rule above we get:</p>\n  \n  \n  \n  <p>\\[\\begin{eqnarray*}\\frac{df}{dx} &#38;=&#38; \\simplify[std]{{m*a}x ^ {m-1} * (1/(2*sqrt(u)))}\\\\\n  \n  &#38;=&#38;\\simplify[std]{{m*a}x^{m-1}/(2*sqrt(u))}\\\\\n  \n  &#38;=&#38; \\simplify[std]{({a*m}x ^ {m-1})/(2*sqrt({a} * x^{m}+{b}))}\n  \n  \\end{eqnarray*}\\]<br/>on replacing $u$ by $\\simplify[std]{{a}x^{m}+{b}}$.</p>\n  \n  \n    ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n        <p>\\[\\simplify[std]{f(x) = sqrt({a} * x^{m}+{b})}\\]</p>\n      <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n      <p>Click on Show steps for more information. You will not lose any marks by doing so.</p>\n      <p>Input all numbers as fractions or integers and not decimals.</p>\n      <p>&nbsp;</p>\n        ", "gaps": [{"notallowed": {"message": "<p>Input all numbers as fractions or integers and not decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 1e-05, "vsetrange": [4.0, 5.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "({a*m}x ^ {m-1})/(2*sqrt({a} * x^{m}+{b}))", "type": "jme"}], "steps": [{"prompt": "\n          \n          \n          <p>The chain rule says that if $f(x)=g(h(x))$ then<br/>\\[\\simplify[std]{f'(x) = h'(x)g'(h(x))}\\]<br/>One way to find $f'(x)$ is to let $u=h(x)$ then we have $f(u)=g(u)$ as a function of $u$.<br/>Then we use the chain rule in the form:<br/>\\[\\frac{df}{dx} = \\frac{du}{dx}\\frac{df}{du}\\]<br/>Once you have worked this out, you replace $u$ by $h(x)$ and your answer is now in terms of $x$.</p>\n          \n          \n            ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "extensions": [], "statement": "<p>Differentiate the following function $f(x)$ using the chain rule.</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(2..8)", "name": "m"}}, "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 - but had to introduce more stringent accuracy constraints - see below.</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>Improved display in prompt.</p>\n                \t\t<p>Added decimal point to forbidden strings and included message not to input decimals.</p>\n                \t\t<p>Increased accuracy threshold to abs diff of 0.00001 and tested the outcomes. OK.</p>\n                \t\t", "description": "\n                \t\t<p>Differentiate</p>\n                \t\t<p>\\[ \\sqrt{a x^m+b})\\]</p>\n                \t\t", "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/"}]}