// Numbas version: finer_feedback_settings {"name": "Chain Rule 06 (non-scaffolded)", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_zWorKza.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_zWorKza.pdf"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Chain Rule 06 (non-scaffolded)", "tags": [], "metadata": {"description": "Instructional \"drill\" exercise to emphasize the method.", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

We use the CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function \"nested\" inside another.

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If $y=f(g(x))$  to find $\\frac{dy}{dx}$ , we need to do two things::

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  1. Substitute $u=g(x)$  so that $y=f(u)$
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  2. Use the Chain Rule:
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\\[ \\large  \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du}  \\]

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Now it is time to get out your paper and pencil and try similar questions without the help. Carry out the same steps, lay it out the same way but you only need to input the final answer.

", "advice": "

We are asked to differentiate:

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\\[    y=\\ln({\\simplify{{CF}x}+\\sin{(x)})}  \\]

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Recognising that this is a \"function of another function\", we need to identify the \"innermost\" of the two functions that are involved and substitute $u$

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Let   $u=\\simplify{{CF}x+sin(x)}$          then          $y=\\ln(u)$

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Now, we need to use the approriate techniques to differentiate each of these, for these functions we need the Power Rule and your Table of Derivatives.:

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Applying this method gives us:

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$\\large \\frac{du}{dx}=\\var{CF}+cos(x)$          and          $ \\large \\frac{dy}{du}= \\frac{1}{u}$

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We now use the Chain Rule formula:

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$ \\large  \\large  \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du}   $

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Make the appropriate substitutions into the formula:

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$ \\large  \\frac{dy}{dx}= (\\var{CF}+cos(x)) \\times \\frac{1}{u}$

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Which simplifies to:

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$ \\large  \\frac{dy}{dx}=\\frac{\\var{CF}+cos(x)}{u}$

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Now, finally, we must remember that $u$ was a variable that we introduced and was not part of the original problem. 

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Replace $u$ from our original substitution to give the final answer:

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$\\large  \\frac{dy}{dx}=\\frac{\\var{CF}+cos(x)}{\\simplify{{CF}x+sin(x)}}$

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Differentiate $   y=\\ln({\\var{CF}x+\\sin{(x)})} $

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$\\large  \\frac{dy}{dx}=$[[0]]

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