Did I write down the derivative product rule correctly for $g(x)=(f(x))^2$ Suppose that $f(4)=5$ and $f'(4)=5$ . Use the product rule to determine the value of $g'(4)$ where $g(x)=(f(x))^2$ 
So I'm writing this problem as:  $g'(x)=(f(x))\frac{d}{dx}f(x)+\frac{d}{dx}f(x)(f(x))$
If anybody can verify that I wrote it down correctly, I would really appreciate it.  

Yes, you've written it correctly. Note that after simplification, the result is $2f(x) f'(x)$, which also agrees with what the chain rule would say in this context.
