Prove that $\frac{d^2y}{dx^2}$ equals $\frac{dy}{dx}×\frac{d}{dy}(\frac{dy}{dx})$
Solution 1:
Recall the chain rule $$ \frac{du}{dx}=\frac{du}{dy}\frac{dy}{dx} $$ and hence $$ \frac{d^2y}{dx^2}=\frac{d\frac{dy}{dx}}{dx}=\frac{d\frac{dy}{dx}}{dy}\frac{dy}{dx}=\frac{dy}{dx}\frac{d}{dy}(\frac{dy}{dx}).$$
Solution 2:
A direct application of chain rule is easiest, unless one is specifically asked to prove this from first principles.
$$ \frac{d^2y}{dx^2} = \frac{d}{dx}(\frac{dy}{dx}) = \frac{d}{dy}(\frac{dy}{dx}) \cdot {\frac{dy}{dx}}$$