// Numbas version: exam_results_page_options {"name": "Milena's copy of Milena's copy of Chain rule ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"a3": {"definition": "random(2..6#1)", "name": "a3", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "a2": {"definition": "random(2..6#1)", "name": "a2", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "a1": {"definition": "random(2..7#1)", "name": "a1", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "a4": {"definition": "random(3..12#1)", "name": "a4", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}}, "statement": "

Differentiate the function:

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

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\$$\\frac{df}{dx}=\$$ [[0]]

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Chain rule

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

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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)=\\var{a1}sin(u)\$$

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

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

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

", "type": "question", "contributors": [{"name": "Milena Venkova", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2169/"}]}]}], "contributors": [{"name": "Milena Venkova", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2169/"}]}