Derivative of derivative squared with respect to same variable

Question

What is the answer, and how should I go forth do derivate something like this. $$\frac d{dx}((\frac {dy}{dx})^2)$$

Answer 1

Let $g=\frac{dy}{dx}$. $$ \frac{d}{dx} \left( \frac{dy}{dx} \right)^2 = \frac{d g^2}{dx} = 2 g\frac{dg}{dx} = 2 \frac{dy}{dx}\frac{d^2y}{dx^2} $$