// Numbas version: finer_feedback_settings
{"name": "Chain rule - binomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Chain rule - binomial", "tags": ["Calculus", "calculus", "chain rule", "checked2015", "derivative of a function of a function", "differentiation", "Differentiation", "function of a function", "Steps", "steps"], "metadata": {"description": "<p>Differentiate $\\displaystyle (ax^m+b)^{n}$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Differentiate the following function $f(x)$ using the chain rule.</p>", "advice": "<p>$\\simplify[std]{f(x) = ({a} * x^{m}+{b})^{n}}$</p>\n<p>The chain rule says that if $f(x)=g(h(x))$ then</p>\n<p>\\[\\simplify[std]{f'(x) = h'(x)g'(h(x))}\\]</p>\n<p>One way to find $f'(x)$ is to let $u=h(x)$ then we have $f(u)=g(u)$ as a function of $u$.</p>\n<p>Then we use the chain rule in the form:</p>\n<p>\\[\\frac{\\mathrm{d}f}{\\mathrm{d}x} = \\frac{\\mathrm{d}u}{\\mathrm{d}x}\\frac{\\mathrm{d}f(u)}{\\mathrm{d}u}\\]</p>\n<p>Once you have worked this out, you replace $u$ by $h(x)$ and your answer is now in terms of $x$.</p>\n<p>For this example, we let $u=\\simplify[std]{{a} * x^{m}+{b}}$ and we have $f(u)=\\simplify[std]{u^{n}}$.</p>\n<p>This gives</p>\n<p>\\begin{align} <br/>\\frac{\\mathrm{d}u}{\\mathrm{d}x} &amp;= \\simplify[std]{{m*a}x ^ {m -1}} \\\\[1em]<br/>\\frac{\\mathrm{d}f(u)}{\\mathrm{d}u} &amp;= \\simplify[std]{{n}u^{n-1}} <br/>\\end{align}</p>\n<p>Hence on substituting into the chain rule above we get:</p>\n<p>\\begin{align}<br/>\\frac{\\mathrm{d}f}{\\mathrm{d}x} &amp;= \\simplify[std]{{m*a}x ^ {m-1} * ({n}*u^{n-1})} \\\\ <br/>&amp;= \\simplify[std]{{m*a*n}x^{m-1}u^{n-1}} \\\\ <br/>&amp;= \\simplify[std]{{m*a*n}x^{m-1}({a}*x^{m}+{b})^{n-1}} <br/>\\end{align}</p>\n<p>on replacing $u$ by $\\simplify[std]{{a}x^{m}+{b}}$.</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "extensions": [], "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(5..9)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "s1", "b", "m", "n"], "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": "<p>\\[\\simplify[std]{f(x) = ({a} * x^{m}+{b})^{n}}\\]</p>\n<p>$\\displaystyle \\frac{\\mathrm{d}f}{\\mathrm{d}x}=$ [[0]]</p>\n<p>Click on <em>Show steps</em> for more information. You will not lose any marks by doing so.</p>", "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": "<p>The chain rule says that if $f(x)=g(h(x))$ then</p>\n<p>\\[\\simplify[std]{f'(x) = h'(x)g'(h(x))}\\]</p>\n<p>One way to find $f'(x)$ is to let $u=h(x)$ then we have $f(u)=g(u)$ as a function of $u$.</p>\n<p>Then we use the chain rule in the form:</p>\n<p>\\[\\frac{\\mathrm{d}f}{\\mathrm{d}x} = \\frac{\\mathrm{d}u}{\\mathrm{d}x}\\frac{\\mathrm{d}f}{\\mathrm{d}u}\\]</p>\n<p>Once you have worked this out, you replace $u$ by $h(x)$ and your answer is now in terms of $x$.</p>"}], "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*m*n}x ^ {m-1} * ({a} * x^{m}+{b})^{n-1}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"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": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}