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{"name": "Differentiation: Quotient Rule 7", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation: Quotient Rule 7", "tags": [], "metadata": {"description": "<p>Calculating the derivative of a function of the form $\\frac{(x^2-a)^n}{x^2+a}$ using the quotient rule.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Find the derivative of \\[ \\simplify{y=(x^2-{a})^{n}/(x^2+{a})}. \\]</p>", "advice": "<p>If we have a function of the form $y=\\tfrac{u(x)}{v(x)}$, to calculate its derivative we need to use the quotient rule:</p>\n<p>\\[ \\dfrac{dy}{dx} = \\dfrac{v(x) \\times \\frac{du}{dx} - u(x) \\times\\frac{dv}{dx}}{[v(x)]^2}\\,.\\]</p>\n<p>This can be split up into steps:</p>\n<ol>\n<li>Identify the functions $u(x)$ and $v(x)$;</li>\n<li>Calculate their derivatives $\\tfrac{du}{dx}$ and $\\tfrac{dv}{dx}$;</li>\n<li>Substitute these into the formula for the quotient rule to obtain an expression for $\\tfrac{dy}{dx}$;</li>\n<li>Simplify $\\tfrac{dy}{dx}$ where possible.</li>\n</ol>\n<p></p>\n<p>Following this process, we must first identify $u(x)$ and $v(x)$.</p>\n<p>As \\[ \\simplify{y=(x^2-{a})^{n}/(x^2+{a})}, \\]</p>\n<p>let \\[ u(x) =&nbsp;\\simplify{(x^2-{a})^{n}} \\quad \\text{and} \\quad v(x)=\\simplify{x^2+{a}}.\\]</p>\n<p>Next, we need to find the derivatives, $\\tfrac{du}{dx}$ and $\\tfrac{dv}{dx}$:</p>\n<p>\\[ \\dfrac{du}{dx} = \\simplify{{2*n}x(x^2-{a})^{n-1}}\\quad \\text{and} \\quad\\dfrac{dv}{dx}=2x.\\]</p>\n<p>Substituting these results into the quotient rule formula we can obtain an expression for $\\tfrac{dy}{dx}$:</p>\n<p>\\[ \\begin{split} \\dfrac{dy}{dx} &amp;\\,= \\dfrac{v(x) \\times \\frac{du}{dx} - u(x) \\times\\frac{dv}{dx}}{[v(x)]^2} \\\\ \\\\&amp;\\,=\\dfrac{(\\simplify{x^2+{a}}) \\times \\simplify{{2*n}x(x^2-{a})^{n-1}} - \\simplify{(x^2-{a})^{n}} \\times \\simplify{2x}}{\\simplify{(x^2+{a})^2}}.&nbsp; \\end{split}\\]</p>\n<p></p>\n<p>{advice}</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "", "templateType": "anything", "can_override": false}, "simplified": {"name": "simplified", "group": "Ungrouped variables", "definition": "\"<p>Simplifying,</p>\\n<p>\\\\[ \\\\begin{split} \\\\dfrac{dy}{dx} &amp;\\\\,= \\\\dfrac{\\\\simplify{{2*n}x(x^2+{a})(x^2-{a})^{n-1}-2x(x^2-{a})^{n}}}{\\\\simplify{(x^2+{a})^2}}&nbsp; \\\\\\\\ \\\\\\\\ &amp;\\\\,= \\\\dfrac{\\\\simplify{2x(x^2-{a})^{n-1}({n}(x^2+{a})-(x^2-{a}))}}{\\\\simplify{(x^2+{a})^2}}\\\\\\\\ \\\\\\\\ &amp;\\\\,= \\\\dfrac{\\\\simplify{2x(x^2-{a})^{n-1}({n-1}x^2+{a*(n+1)})}}{\\\\simplify{(x^2+{a})^2}}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "simplify": {"name": "simplify", "group": "Ungrouped variables", "definition": "\"<p>Simplifying,</p>\\n<p>\\\\[ \\\\begin{split} \\\\dfrac{dy}{dx} &amp;\\\\,= \\\\dfrac{\\\\simplify{{2*n}x(x^2+{a})(x^2-{a})^{n-1}-2x(x^2-{a})^{n}}}{\\\\simplify{(x^2+{a})^2}}&nbsp; \\\\\\\\ \\\\\\\\ &amp;\\\\,= \\\\dfrac{\\\\simplify{2x(x^2-{a})^{n-1}({n}(x^2+{a})-(x^2-{a}))}}{\\\\simplify{(x^2+{a})^2}}\\\\\\\\ \\\\\\\\ &amp;\\\\,= \\\\dfrac{\\\\simplify{2x(x^2-{a})^{n-1}({n-1}x^2+{a*(n+1)})}}{\\\\simplify{(x^2+{a})^2}}\\\\\\\\ \\\\\\\\ &amp;\\\\,= \\\\dfrac{\\\\simplify{{2*simp}x(x^2-{a})^{n-1}({(n-1)/simp}x^2+{(a*(n+1))/simp})}}{\\\\simplify{(x^2+{a})^2}}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "advice": {"name": "advice", "group": "Ungrouped variables", "definition": "if(simp=1,'{simplified}','{simplify}')", "description": "", "templateType": "anything", "can_override": false}, "simp": {"name": "simp", "group": "Ungrouped variables", "definition": "gcd(n-1,a*(n+1))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "n", "simplified", "simplify", "advice", "simp"], "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>$\\dfrac{dy}{dx}=$[[0]]&nbsp;</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": "{2*simp}*x*(x^2-{a})^{n-1}({(n-1)/simp}*x^2+{(a*(n+1))/simp})/(x^2+{a})^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "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/"}]}