simplest possible way to prove chain rule for second derivative

Question

Given $\phi=\phi(u)$ and $u=u(x)$

What is the simplest possible way to prove $$ \frac{d^2\phi}{dx^2}=\bigg(\frac{du}{dx}\bigg)^2.\frac{d^2\phi}{du^2}+\frac{d^2u}{dx^2}.\frac{d\phi}{du} $$

Answer 1

by repeated application of the chain rule: $$ \frac{d^2\phi}{dx^2}=\frac{d}{dx}\big(\frac{d\phi}{dx}\big)=\frac{d}{dx}\big(\frac{d\phi}{du}\frac{du}{dx}\big)=\frac{d}{dx}\big(\frac{d\phi}{du}\big)\frac{du}{dx}+\frac{d\phi}{du}\frac{d^2u}{dx^2}=\frac{du}{dx}\frac{d}{du}\big(\frac{d\phi}{du}\big)\frac{du}{dx}+\frac{d\phi}{du}\frac{d^2u}{dx^2}=\frac{d^2\phi}{du^2}\big(\frac{du}{dx}\big)^2+\frac{d\phi}{du}\frac{d^2u}{dx^2} $$

where $\frac{d}{dx}=\frac{du}{dx}\frac{d}{du}$

Answer 2

Have you tried just iterating the chain rule for the first derivative? It's the first thing that comes to mind and a simple computation does the trick.

Hope that helps,