Find the equivalent formula for second derivative

Question

Find the equivalent formula(that still use Leibniz's notation) of $$\frac {d^2f(u)}{dx^2}$$using th chain rule. $u$ is a function of $x$.

Answer 1

$$\begin{aligned}\frac{d^2f}{dx^2}&=\frac{d}{dx}\left(\frac{df}{du}\frac{du}{dx}\right)\\ &=\frac{df}{du}\frac{d^2u}{dx^2}+\frac{d}{dx}\left(\frac{df}{du}\right)\frac{du}{dx}\\ &=\frac{df}{du}\frac{d^2u}{dx^2}+\left(\frac{d^2f}{du^2}\frac{du}{dx}\right)\frac{du}{dx}\\ &=\frac{df}{du}\frac{d^2u}{dx^2}+\frac{d^2f}{du^2}\left(\frac{du}{dx}\right)^2 \end{aligned}$$

Answer 2

You can use Leibniz rule: $$\frac{d}{dx}\Big(\frac{d}{dx}f(u(x)) \Big) = \frac{d}{dx}\Big(\frac{d}{du}f(u)\frac{d}{dx}u(x) \Big) = \frac{d^2}{du^2}f(u)\Big(\frac{d}{dx}u(x)\Big)^2+\frac{d}{du}f(u)\frac{d^2}{dx^2}u(x).$$