// Numbas version: exam_results_page_options
{"name": "Harry's copy of cormac's copy of Chain rule - exponential of polynomial, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Harry's copy of cormac's copy of Chain rule - exponential of polynomial, ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "variablesTest": {"maxRuns": 100, "condition": ""}, "functions": {}, "variables": {"s2": {"name": "s2", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1,-1)"}, "s1": {"name": "s1", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1,-1)"}, "c": {"name": "c", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "s2*random(1..9)"}, "a": {"name": "a", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(2..9)"}, "m": {"name": "m", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(3..4)"}, "b": {"name": "b", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "s1*random(1..9)"}}, "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["a", "c", "b", "s2", "s1", "m"], "statement": "<p>Differentiate the following function $f(x)$ using the chain rule.</p>", "variable_groups": [], "extensions": [], "parts": [{"variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "type": "gapfill", "prompt": "<p>\\[\\simplify[std]{f(x) = e^({a}x^{m} +{b}x^2+{c})}\\]</p>\n<p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>", "gaps": [{"showFeedbackIcon": false, "type": "jme", "checkingaccuracy": 0.001, "expectedvariablenames": [], "checkingtype": "absdiff", "vsetrangepoints": 5, "showCorrectAnswer": true, "vsetrange": [0, 1], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "answersimplification": "std", "scripts": {}, "marks": 3, "answer": "({m*a}x^{m-1}+{2*b}x)*e^({a}x^{m} +{b}x^2+{c})", "checkvariablenames": false, "showpreview": true}], "marks": 0, "variableReplacements": []}], "tags": [], "advice": "\n\t    \n\t    \n\t    <p>$\\simplify[std]{f(x) = e^({a}x^{m} +{b}x^2+{c})}$<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 $u=\\simplify[std]{{a}x^{m} +{b}x^2+{c}}$ and we have $f(u)=\\simplify[std]{e^u}$.<br/>This gives<br/>\\[\\begin{eqnarray*}\\frac{du}{dx} &#38;=&#38; \\simplify[std]{{a*m}x^{m-1} +{2*b}x}\\\\\n\t    \n\t    \\frac{df(u)}{du} &#38;=&#38; \\simplify[std]{e^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}x^{m-1} +{2*b}x) * (e^u)}\\\\\n\t    \n\t    &#38;=&#38; \\simplify[std]{({a*m}x^{m-1} +{2*b}x)*e^({a}x^{m} +{b}x^2+{c})}\n\t    \n\t    \\end{eqnarray*}\\]<br/>on replacing $u$ by $\\simplify[std]{{a}x^{m} +{b}x^2+{c}}$.</p>\n\t    \n\t    \n\t", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Differentiate $\\displaystyle e^{ax^{m} +bx^2+c}$</p>"}, "type": "question", "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}]}], "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}