// Numbas version: exam_results_page_options {"name": "Chain rule - trig and quadratic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Chain rule - trig and quadratic", "tags": ["calculus", "Calculus", "Chain rule", "chain rule", "checked2015", "derivative of a function of a function", "differentiation", "Differentiation", "function of a function", "steps", "Steps"], "metadata": {"description": "

Differentiate $\\displaystyle \\cos(e^{ax}+bx^2+c)$

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "

$\\simplify[std]{f(x) = cos(e^({a}x) +{b}x^2+{c})}$

The chain rule says that if $f(x)=g(h(x))$ then

\\[\\simplify[std]{f'(x) = h'(x)g'(h(x))}\\]

One way to find $f'(x)$ is to let $u=h(x)$ then we have $f(u)=g(u)$ as a function of $u$.

Then we use the chain rule in the form:

\\[\\frac{df}{dx} = \\frac{du}{dx}\\frac{df(u)}{du}\\]

Once you have worked this out, you replace $u$ by $h(x)$ and your answer is now in terms of $x$.

For this example, we let $u=\\simplify[std]{e^({a}x) +{b}x^2+{c}}$ and we have $f(u)=\\simplify[std]{cos(u)}$.

This gives

\\begin{align}
\\frac{\\mathrm{d}u}{\\mathrm{d}x} &= \\simplify[std]{{a}e^({a}x) +{2*b}x} \\\\[1em]
\\frac{\\mathrm{d}f(u)}{\\mathrm{d}u} &= \\simplify[std]{-sin(u)}
\\end{align}

Hence on substituting into the chain rule above we get:

\\begin{align}
\\frac{\\mathrm{d}f}{\\mathrm{d}x} &= \\simplify[std]{({a}e^({a}x) +{2*b}x) * (-sin(u))} \\\\
&= \\simplify[std]{-({a}e^({a}x) +{2*b}x)*sin(e^({a}x) +{b}x^2+{c})}
\\end{align}

on replacing $u$ by $\\simplify[std]{e^({a}x) +{b}x^2+{c}}$.

", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "s2": {"name": "s2", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "s2", "c", "b", "s1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Differentiate the following function $f(x)$ using the chain rule.

\\[\\simplify[std]{f(x) = cos(e^({a}x) +{b}x^2+{c})}\\]

$\\displaystyle \\frac{\\mathrm{d}f}{\\mathrm{d}x}=$ [[0]]

", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

The chain rule says that if $f(x)=g(h(x))$ then

\\[\\simplify[std]{f'(x) = h'(x)g'(h(x))}\\]

One way to find $f'(x)$ is to let $u=h(x)$ then we have $f(u)=g(u)$ as a function of $u$.

Then we use the chain rule in the form:

\\[\\frac{\\mathrm{d}f}{\\mathrm{d}x} = \\frac{\\mathrm{d}u}{\\mathrm{d}x}\\frac{\\mathrm{d}f}{\\mathrm{d}u}\\]

Once you have worked this out, you replace $u$ by $h(x)$ and your answer is now in terms of $x$.

"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 3, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-({a}e^({a}x)+{2*b}x)*sin(e^({a}x) +{b}x^2+{c})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}