// Numbas version: exam_results_page_options {"name": "Katy's copy of Katy's copy of Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "name": "Katy's copy of Katy's copy of Product rule", "tags": [], "parts": [{"gaps": [{"expectedVariableNames": [], "variableReplacements": [], "checkingAccuracy": 0.001, "vsetRange": [0, 1], "unitTests": [], "showPreview": true, "answer": "{a}{g}x^{a-1}({c}x^{d}+{b}) + {g}x^{a} ({c}{d}x^{d-1}+{b})", "marks": "3", "customMarkingAlgorithm": "", "vsetRangePoints": 5, "scripts": {}, "checkVariableNames": false, "showFeedbackIcon": true, "failureRate": 1, "type": "jme", "checkingType": "absdiff", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst"}], "prompt": "

\\(\\frac{dy}{dx}=\\)[[0]]

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\\(y= \\var{g}x^{\\var{a}} \\times (\\var{c}x^{\\var{d}} + \\var{b})\\)

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Recall the product rule if \\(y=u \\times v\\) where \\(u\\) and \\(v\\) are both functions of \\(x\\) then

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\\(\\frac{dy}{dx}=v \\times \\frac{du}{dx}+u \\times \\frac{dv}{dx}\\)

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let \\( u = \\var{g}x^{\\var{a}}\\)  and  \\(v=\\var{c}x^{\\var{d}} + \\var{b}\\)

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\\(\\frac{du}{dx}=\\var{a}\\var{g}x^\\var{a-1}\\)  and  \\(\\frac{dv}{dx}=\\var{c}\\var{d}x^{\\var{d-1}}\\)

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Substituting into \\(\\frac{dy}{dx}=v \\times \\frac{du}{dx}+u \\times \\frac{dv}{dx}\\) gives

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

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Which tidies up to \\(\\frac{dy}{dx}= \\var{a}\\var{g}x^\\var{a-1}(\\var{c}x^{\\var{d}} + \\var{b})  +  \\var{g}x^\\var{a} (\\var{c}\\var{d}x^{\\var{d-1}} + \\var{b})\\)

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Product rule

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Differentiate the following using the product rule:

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\\(y=\\var{g}x^{\\var{a}} \\times (\\var{c}x^{\\var{d}} + \\var{b})\\)

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