# What is the derivative of the following?

If the derivative of $y=g(x)$ equals $12$ when $x=2$,

what is the derivative of $y=g(\frac13 x+1)$ when $x=3$?

I really don't even know where to start, any suggestions would be great!

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You are given that $g'(2)=12$. If you let $h(x)=\tfrac{1}{3}x+1$ and $f(x)=g(h(x))$ then you are being asked for $f'(3)$. Use the chain rule, $$f'(3)=g'(h(3))\cdot h'(3)=g'(2)h'(3)=12\cdot\frac{1}{3}=4.$$

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oh ok! I was not connecting that I could use g'(2) to help me solve g'(h(3)). Thank you! –  user56852 Feb 4 '13 at 1:31

Suggestion: Use the chain rule.

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yes but I dont even know what I am using the chain rule on! Do i need to plug something into something else? –  user56852 Feb 4 '13 at 1:17
@user56852: The chain rule is for situations where you are differentiating a function composition, something like $g(f(x))$. You have such an expression in your problem. What is $f(x)$? –  Jonas Meyer Feb 4 '13 at 1:18
f(x) would be ((1/3)x)+1. so I know I need to derive g(f(x)) and then multiply that by the derivative of f(x). How do I derive it when I am not actually given a g(x) function? –  user56852 Feb 4 '13 at 1:24
@user56852: You are given some information about the derivative of $g$, and it is exactly what you will need to solve this problem by using the chain rule. It is OK that you don't see in advance exactly where it will end up. Just try applying the chain rule as best as you can, starting with the general formula for what the derivative of $g(f(x))$ is in terms of $f$ and $g$, and then plugging in everything possible based on the given information. –  Jonas Meyer Feb 4 '13 at 1:26
What you stated, "derive $g(f(x))$ and then multiply that by the derivative of $f(x)$", is incorrect, because what you want is the derivative of $g(f(x))$; finding the latter involves multiplying something by the derivative of $f(x)$, but not the thing you are finding. –  Jonas Meyer Feb 4 '13 at 1:27