Q:

If $g'(x)$ is constant, what is the fifth derivative of $f(g(x))$?

I just need some clarification about how to solve and approach this problem, we did it in a Calc 1474 class but I didn't understand how to approach and solve it. 

A:

You can use the chain rule $h(x)=f(g(x))$ implies that $h'(x)=f'(g(x))g'(x)$.
Then, chain rule and product rule: $h''(x)=f''(g(x))(g'(x))^2+f'(g(x))g''(x)$. Since $g'$ is constant then $g''(x)=0$ for all $x$. Therefore $h''(x)=f''(g(x))(g'(x))^2$.
You can continue in the same way (notice that whenever you try to compute the second term in the product rule you get a $g''$ which is zero and the term dissapears) to get that:
\begin{align}h'''(x)&=f'''(g(x))(g'(x))^3\\
h^{(iv)}(x)&=f^{(iv)}(g(x))(g'(x))^4\\
h^{(v)}(x)&=f^{(v)}(g(x))(g'(x))^5\end{align}

