Reason behind notation of $\frac{d^{2}y}{dx^{2}}$

Question

Why is the notation the way it is? Is there any history behind it, or any logic which I am not aware of?

Answer 1

Indeed this notation is not very natural. It is meant to be a shorthand of $\frac d{dx} \frac{dy}{dx}$. But since differentiation of quotients is not equal to the quotient of differentiation, a more logical notation might be $$ \frac d{dx}\frac{dy}{dx} = \frac{\frac d{dx} dy}{dx} - \frac{dy \frac d{dx} dx}{(dx)^2}.$$ But this is just too long to write out. We regard $dx$ as a single entity, so $dx^2$ is short for $(dx)^2$ (i.e. $d$ has higher precedence than exponentiation, you can revise PEMDAS to PDEMDAS). And the $d^2 y$ is just there to remind you that it was $d \ dy$ on the numerator.

Answer 2

A short answer to understand why the notation

$$ \frac{dy^2}{dx^2} $$

would be bad to indicate the second derivative (of $y$ with respect to $x$) is that dimensions don't agree. The second derivative has dimensions of $[y]/[x]^2$ while the notation above has dimensions $[y]^2/[x]^2$ (which is wrong).

For a more detailed answer you can look at this question as noted by @Bumblebee.