Solutions

We have that $g(x)=f(-x)=f(x)$. Hence,

$$\begin{align*} f'(x)&=g'(x)=\lim_{h\to 0}\frac{g(x+h)-g(x)}{h} \\ &=\lim_{h\to 0}\frac{f(-x-h)-f(-x)}{h} \\ &=\lim_{k\to 0}\frac{f(-x+k)-f(-x)}{-k} \\ &=-\lim_{k\to 0}\frac{f(-x+k)-f(-x)}{k} \\ &=-f'(-x). \end{align*}$$