**Re: Finding The Derivative of a composition of a function...**

Use the chain rule to state:

\(\displaystyle g'(x)=f'\(f(x)\)f'(x)\)

hence:

\(\displaystyle g'(1)=f'\(f(1)\)f'(1)\)

So, we need to compute, using the product and chain rules:

\(\displaystyle f'(x)=x\(e^{x^2}\cdot2x\)+(1)\(e^{x^2}\)=e^{x^2}\(2x^2+1\)\)

We now need:

\(\displaystyle f(1)=e\)

\(\displaystyle f'\(f(1)\)=f'(e)=e^{e^2}\(2e^2+1\)\)

\(\displaystyle f'(1)=3e\)

and so:

\(\displaystyle g'(1)=\(e^{e^2}\(2e^2+1\)\)\(3e\)=3e^{e^2+1}\(2e^2+1\)\)