// Numbas version: exam_results_page_options
{"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", "logarithm laws", "logarithms", "steps"], "advice": "\n  \n  \n  <p>$\\simplify[std]{f(x) = ln(({a}x+{b})^{m})}$<br/>First note that we can simplify this by using the rule that $\\simplify[std]{ln(a^r)=r*ln(a)}$.<br/>Hence $\\simplify[std]{f(x) = ln(({a}x+{b})^{m})={m}ln({a}x+{b})}$<br/>So we need to differentiate $\\simplify[std]{ln({a}x+{b})}$<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 $u=\\simplify[std]{{a}x +{b}}$ and we have $f(u)=\\simplify[std]{{m}*ln(u)}$.<br/>This gives<br/>\\[\\begin{eqnarray*}\\frac{du}{dx} &#38;=&#38; \\simplify[std]{{a}}\\\\\n  \n  \\frac{df(u)}{du} &#38;=&#38; \\simplify[std]{{m}/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]{({a}) * ({m}/u)}\\\\\n  \n  &#38;=&#38; \\simplify[std]{{a*m}/({a}x+{b})}\n  \n  \\end{eqnarray*}\\]<br/>on replacing $u$ by $\\simplify[std]{{a}x+{b}}$.</p>\n  \n  \n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n      <p>\\[\\simplify[std]{f(x) = ln(({a}x+{b})^{m})}\\]</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      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [5.0, 6.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "({m*a})/({a}x+{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"}, "c": {"definition": "s2*random(1..9)", "name": "c"}, "b": {"definition": "s1*random(1..9)", "name": "b"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "m": {"definition": "random(3..9)", "name": "m"}}, "metadata": {"notes": "\n                            \t\t<p>&nbsp;</p>\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>Improved display in prompt.</p>\n                            \t\t<p>Checking range chosen so that the denominator of the result is never 0.</p>\n                            \t\t<p>&nbsp;</p>\n                            \t\t", "description": "<p>Differentiate $\\displaystyle \\ln((ax+b)^{m})$</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/"}]}