I know the second derivative of y wrt x: $\frac{d^2y}{dx^2}$ means $\frac{d}{dx}(\frac{dy}{dx})$, but is there a mathematical reason you square the $d$ in the numerator but the $x$ in the denominator?
I've wondered if it's because the $d$ in the denominator represents some arbitrary infinitely tiny amount, and the $d$ in the numerator is that same $d$, only squared to account for the second derivative.
Does that make sense, and/or am I missing something significant about derivatives?
$\frac{d}{dx}$ is a linear operator on the space of differentiable functions. When we write $\frac{dy}{dx}$, we are really saying $\frac{d}{dx}(y)$, where $y$ represents some function of $x$. Doing this twice, we get $$\frac{d}{dx}\frac{d}{dx}(y)=\left(\frac{d}{dx}\right)^2(y)=\frac{d^2y}{dx^2}$$
This is just a widely accepted abuse of notation. If we didn't abuse notation, we would instead write $\frac{d^2 y}{(dx)^2}$.
It's important to keep in mind that this notation of $\frac{dy}{dx}$, called Leibniz notation, is chosen to help you remember certain formulas (the chain rule) but isn't really rigorous on its own. You can certainly get yourself into some trouble by just pushing these symbols around like you would fractions!
