// Numbas version: finer_feedback_settings
{"name": "Differentiation (Harder powers - negative and fractions) 4b", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation (Harder powers - negative and fractions) 4b", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>If we have a function of the form $y=f(g(x))$, sometimes described as a function of a function, to calculate its derivative we need to use the chain rule:</p>\n<p>\\[ \\frac{dy}{dx} = \\frac{du}{dx} \\times \\frac{dy}{du}.\\]</p>\n<p></p>\n<p>This can be split up into stages:</p>\n<ul>\n<li>Let $u=g(x)$;</li>\n<li>Rewrite $y$ in terms of $u$, such that $y=f(u)$;</li>\n<li>Calculate $\\frac{du}{dx}$ and $\\frac{dy}{du}$;</li>\n<li>Write $\\frac{dy}{dx}$ as a product of&nbsp;$\\frac{du}{dx}$ and $\\frac{dy}{du}$;</li>\n<li>Make sure $\\frac{dy}{dx}$ is&nbsp;<em>only</em> in terms of $x$. Ensure any $u$ terms have been replaced using the initial substitution.</li>\n</ul>\n<p></p>\n<p>Calculate the derivative of $y=\\simplify[all,fractionNumbers]{{k}({a}*x^{m}+{b})^{n}}$.</p>\n<p></p>", "advice": "<p>Following the process of calculating the derivative of a function using the chain rule, we must first identify $g(x)$. Since the function is of the form $y=f(g(x))$, we are looking for the 'inner' function:</p>\n<p>So, for $y=\\simplify[all,fractionNumbers]{{k}({a}*x^{m}+{b})^{n}}$, \\[g(x)=\\simplify[all, fractionNumbers, unitFactor]{{a}x^{m}+{b}}.\\]</p>\n<p>If we now set $u=g(x)$, we can rewrite $y$ in terms of $u$ such that $y=f(u)$:</p>\n<p>\\[y=\\simplify[all, fractionNumbers,unitFactor]{{k}u^{n}}.\\]</p>\n<p>Next, we calculate the two derivatives $\\frac{du}{dx}$ and $\\frac{dy}{du}$:</p>\n<p>\\[\\frac{du}{dx}=\\simplify[all,fractionNumbers]{{a*m}x^{m-1}}, \\quad \\frac{dy}{du}=\\simplify[all, fractionNumbers, unitFactor]{{k*n}*u^{n-1}}.\\]</p>\n<p>Plugging these into the chain rule:</p>\n<p>\\[ \\begin{split} \\frac{dy}{dx} &amp;= \\frac{du}{dx} \\times \\frac{dy}{du}, \\\\&amp;=\\simplify[all,fractionNumbers]{{a*m}x^{m-1}} \\times\\simplify[all, fractionNumbers, unitFactor]{{k*n}u^{n-1}}, \\\\ &amp;=\\simplify[all, fractionNumbers, unitFactor]{{k*n*a*m}*x^{m-1}u^{n-1}}. \\end{split} \\]</p>\n<p>Finally, we need to express $\\frac{dy}{dx}$&nbsp;<em>only&nbsp;</em>in terms of $x$, so we must replace the $u$ terms using the initial substitution $u=\\simplify[all, fractionNumbers, unitFactor]{{a}x^{m}+{b}}$:</p>\n<p>\\[ \\frac{dy}{dx} =&nbsp;\\simplify[all, fractionNumbers, unitFactor]{{k*n*a*m}*x^{m-1}({a}x^{m}+{b})^{n-1}}.\\]</p>\n<p></p>", "rulesets": {}, "extensions": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..7)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "n_1/n_2", "description": "", "templateType": "anything"}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything"}, "n_1": {"name": "n_1", "group": "Ungrouped variables", "definition": "random(-8..8)", "description": "", "templateType": "anything"}, "n_2": {"name": "n_2", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "k*n*a*m<100 and k*n*a*m>-100 and gcd(n_1,n_2)=1", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "n", "k", "m", "n_1", "n_2"], "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>$\\frac{dy}{dx}=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a*n*k}*x^{m-1}*({a}x^{m}+{b})^{n-1}", "answerSimplification": "fractionNumbers, basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}