Answer
See the explanation.
Work Step by Step
Let $F=f\circ g$.
Define $u=g(x)$. Then, $F=f(u)$.
Find $F'(x)$:
$F'(x)=\frac{dF}{dx}$ (Use the Chain Rule)
$=\frac{dF}{du}\cdot \frac{du}{dx}$
$=f'(u)\cdot g'(x)$
$=f'(g(x))\cdot g'(x)$
Find $F''(x)$:
$F''(x)=\frac{dF'}{dx}$
$=\frac{d}{dx}(f'(g(x))\cdot g'(x))$ (Use the Product Rule)
$=\frac{d}{dx}(f'(g(x))\cdot g'(x)+f'(g(x))\cdot \frac{d}{dx}(g'(x))$
$=\frac{d}{dx}(f'(u))\cdot g'(x)+f'(g(x))\cdot g''(x)$ (Use the Chain Rule)
$=\frac{df'}{du}\cdot \frac{du}{dx}\cdot g'(x)+f'(g(x))\cdot g''(x)$
$=f''(u)\cdot g'(x)\cdot g'(x)+f'(g(x))\cdot g''(x)$
$=f''(g(x))\cdot [g'(x)]^2+f'(g(x))\cdot g''(x)$
