What is the notation for the second derivative of a function?

Question

Is the notation for the second derivative of a function $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}$ or $\frac{\mathrm{d}^2y}{\mathrm{d^2}x}$

Answer 1

We think of the derivative as an "operator" that acts on a function. One symbol for this operator is $\frac{d}{dx}.$ If you apply this operator twice, you get

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

So that's why only the $d$ is squared on top, but the whole $dx$ is squared on the bottom.