What does the denominator in the second derivative mean?

Question

This occurred to me a few days ago.

We know that the derivative of a function $y=f(x)$ is $\frac{dy}{dx}$. This is because it represents how $y$ changes with $x$, which is the rate of change of $y$, or more specifically, the gradient of a function.

Then the second derivative is the rate of change of rate of change, or the rate of change of gradient. Since a general rate of change is $\frac{d}{dx}$, the second derivative is $(\frac{d}{dx})(\frac{dy}{dx})$. Thus, the expanded form is $\frac{d^2y}{dx^2}$.

My question is, is the denominator $d(x)^2$ or is it $(dx)^2$? Surely, it would be the latter, because when you expand $(\frac{d}{dx})(\frac{dy}{dx})$, the $(dx)(dx)$ would become $(dx)^2$. But then why is it never written with brackets? I'm sure that would confuse some people and I only realised it myself when I started thinking about the second derivative properly, in terms of what it actually means.

Answer 1

You are completely correct. When we write $\frac{d^2y}{dx^2}$, we really mean to write $\frac{d^2y}{(dx)^2}$, but those parentheses make the denominator difficult to read and write. Mathematicians accept the form $\frac{d^2y}{dx^2}$ without question, because it is not exactly an algebraic expression (although sometimes it can be treated as one), rather it is a notation that represents the concept of finding the rate of change of the rate of change.

Answer 2

You're right that it should really be $(dx)^2$ (if you were asking "why should that be?", see my answer here).

I suspect the brackets/parentheses aren't written because Leibniz himself didn't write them when he first presented the notation and everyone just followed his lead.