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{"name": "Chain rule - log of binomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "name": "c"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "a"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..9)", "description": "", "name": "m"}}, "ungrouped_variables": ["a", "c", "b", "s2", "s1", "m"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Chain rule - log of binomial", "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "({m*a})/({a}x+{b})", "showCorrectAnswer": true, "vsetrange": [5, 6], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t<p>\\[\\simplify[std]{f(x) = ln(({a}x+{b})^{m})}\\]</p>\n\t\t\t        <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n\t\t\t        <p>Click on Show steps for more information. You will not lose any marks by doing so.</p>\n\t\t\t", "steps": [{"type": "information", "prompt": "\n\t\t\t\t\t            \n\t\t\t\t\t            \n\t\t\t\t\t            <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\t\t\t\t\t            \n\t\t\t\t\t            \n\t\t\t\t\t", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Differentiate the following function $f(x)$ using the chain rule.</p>", "tags": ["Calculus", "MAS1601", "Steps", "chain rule", "checked2015", "derivative of a function of a function", "differentiation", "function of a function", "logarithm laws", "logarithms"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "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", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Differentiate $\\displaystyle \\ln((ax+b)^{m})$</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t    \n\t    \n\t    <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\t    \n\t    \n\t    \n\t    <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\t    \n\t    \\frac{df(u)}{du} &#38;=&#38; \\simplify[std]{{m}/u} \\end{eqnarray*}\\]</p>\n\t    \n\t    \n\t    \n\t    <p>Hence on substituting into the chain rule above we get:</p>\n\t    \n\t    \n\t    \n\t    <p>\\[\\begin{eqnarray*}\\frac{df}{dx} &#38;=&#38; \\simplify[std]{({a}) * ({m}/u)}\\\\\n\t    \n\t    &#38;=&#38; \\simplify[std]{{a*m}/({a}x+{b})}\n\t    \n\t    \\end{eqnarray*}\\]<br/>on replacing $u$ by $\\simplify[std]{{a}x+{b}}$.</p>\n\t    \n\t    \n\t", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}