// Numbas version: exam_results_page_options {"name": "Katy's copy of Quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"type": "gapfill", "customMarkingAlgorithm": "", "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "sortAnswers": false, "showFeedbackIcon": true, "marks": 0, "unitTests": [], "gaps": [{"expectedVariableNames": [], "type": "jme", "unitTests": [], "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "checkingType": "absdiff", "showFeedbackIcon": true, "showPreview": true, "vsetRange": [0, 1], "checkVariableNames": false, "failureRate": 1, "marks": "3", "answer": "({b}{g}x^{b-1}({c}x^{d} + {f})-{d}{c}x^{d-1}({a} + {g}x^{b}))/({c}x^{d} +{f})^2", "vsetRangePoints": 5, "customMarkingAlgorithm": "", "checkingAccuracy": 0.001}], "prompt": "

\$$\\frac{df}{dx} = \$$ [[0]]

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\$$f(x)=\\frac{\\var{a} +\\var{g}x^{\\var{b}}}{\\var{c}x^{\\var{d}} +\\var{f}}\$$

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Recall the quotient rule: if \$$y=\\frac{u}{v}\$$ where \$$u\$$ and \$$v\$$ are both functions of \$$x\$$ then

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\$$\\frac{dy}{dx}=\\frac{v\\frac{du}{dx}-u\\frac{dv}{dx}}{v^2}\$$

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Let   \$$u=\\var{a} +\\var{g}x^{\\var{b}}\$$   and   \$$v=\\var{c}x^{\\var{d}} +\\var{f}\$$

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then   \$$\\frac{du}{dx}=\\var{g}\\var{b}x^{\\var{b-1}}\$$   and   \$$\\frac{dv}{dx}=\\var{d}\\var{c}x^{\\var{d-1}}\$$

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Putting these results together as shown in the rule gives:

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\$$\\frac{df}{dx}=\\frac{(\\var{c}x^{\\var{d}} +\\var{f}) \\times \\var{g}\\var{b}x^{\\var{b-1}} - (\\var{a} +\\var{g}x^{\\var{b}}) \\times \\var{d}\\var{c}x^{\\var{d-1}}}{(\\var{c}x^{\\var{d}} +\\var{f})^2}\$$

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\$$\\frac{df}{dx}=\\frac{\\var{g}\\var{b}x^{\\var{b-1}}(\\var{c}x^{\\var{d}} +\\var{f}) - \\var{d}\\var{c}x^{\\var{d-1}}(\\var{a} +\\var{g}x^{\\var{b}}) }{(\\var{c}x^{\\var{d}} +\\var{f})^2}\$$

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Differentiate the function:

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\$$f(x)=\\frac{\\var{a} + \\var{g}x^{\\var{b}}}{\\var{c}x^{\\var{d}} + \\var{f}}\$$

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Quotient rule

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