$f(x)=3x^2 + g(x^2) $ and x=2 is a critical point

couldnt solve this at all, how can I derivate the [g(x^2)]'?

1 Answer

It is not clear what this question is asking, but I guess you want to compute $g'(4)$ given that $x=2$ is a critical point. 

Let 
\[f(x)=3x^2+g(x^2).\]
Then using chain rule 
\[f'(x)=6x+2x g'(x^2).\]
Substituting $x=2$ we get 
\[0=f'(2)=6\cdot (2)+2\cdot 2g'(4)\]
\[\Rightarrow    12+4g'(4)=0   \Rightarrow g'(4)=-3\]

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