// Numbas version: exam_results_page_options {"name": "Katy's copy of Chain rule 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a1", "a2", "a3", "a4"], "name": "Katy's copy of Chain rule 4", "parts": [{"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "showCorrectAnswer": true, "sortAnswers": false, "showFeedbackIcon": true, "type": "gapfill", "unitTests": [], "variableReplacements": [], "scripts": {}, "gaps": [{"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "failureRate": 1, "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "jme", "answer": "({a1}*{a2}*{a3}x^{{a3}-1})({a2}x^{{a3}}+{a4})^{{a1}-1}", "vsetRange": [0, 1], "checkVariableNames": false, "showPreview": true, "checkingAccuracy": 0.001, "unitTests": [], "variableReplacements": [], "scripts": {}, "expectedVariableNames": [], "vsetRangePoints": 5, "checkingType": "absdiff", "customMarkingAlgorithm": ""}], "prompt": "

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

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\\(f(x)=({\\var{a2}x^{\\var{a3}}+\\var{a4}})^\\var{a1}\\)

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Recall the chain rule:   \\(\\frac{df}{dx}=\\frac{df}{du}.\\frac{du}{dx}\\)

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let \\(u=\\var{a2}x^{\\var{a3}}+\\var{a4}\\)    then   \\(f(x)=u^\\var{a1}\\)

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\\(\\frac{df}{du}=\\var{a1}u^{\\var{a1}-1}\\)  and  \\(\\frac{du}{dx}=\\var{a3}*\\var{a2}x^{\\var{a3}-1}\\)

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\\(\\frac{df}{dx}=\\var{a1}u^{\\simplify{{a1}-1}}.\\simplify{{a2}*{a3}x^{{a3}-1}}\\)

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\\(\\frac{df}{dx}=\\simplify{{a1}*{a2}*{a3}x^{{a3}-1}}({\\var{a2}x^{\\var{a3}}+\\var{a4}})^{\\simplify{{a1}-1}}\\)

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Chain rule

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Differentiate the function:

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\\(f(x)=({\\var{a2}x^{\\var{a3}}+\\var{a4}})^\\var{a1}\\)

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