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I am new to linear approximation. I understand how to linearly approximate with $f(x)$ but I am confused about how to use a linear approximation for composite functions. I know the formula being $f(g(x))=f(g(a))+f'(g(a))g(x)$. I don't understand why we don't take the derivative of $g(x)$ as we do in the chain rule. Also, why is it that we don't multiply the derivative of $f(g(x))$ with the change in $x$? Thank you.

Samuel Adrian Antz
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Voltage crayon 24
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    You should take the derivative of $g$ as well. The (affine) approximation of the composition is $f(g(a)) + f'(g(a)) g'(a) (x-a)$. See Intuitive Proof of the Multivariable Chain Rule for some more details about motivating the chain rule and how it relates to linear approximations. The answer there is written in the multivariable situation but you can ignore that and assume it's all single-variable if you like. – peek-a-boo Jul 08 '22 at 19:41
  • @peek-a-boo, So for example, if I have the function $e^x^4$ for x near 0, what would the linear approximation be? Can you also show your steps, please? Thank you – Voltage crayon 24 Jul 08 '22 at 21:35
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    I've already given you the general formula. Here, $f(y)=e^y$ and $g(x)=x^4$, and $a=0$. Plug this into the above expression and see what you get. The real question is where you got the incorrect formula $f(g(x))=f(g(a))+f'(g(a))g(x)$ from. – peek-a-boo Jul 08 '22 at 21:40
  • @peek-a-boo Okay, thanks for your help. – Voltage crayon 24 Jul 08 '22 at 21:54

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