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?

The notation is very natural: $\displaystyle \frac{d}{dx} \frac{d}{dx} =\frac{d^2}{dx^2}$ — azif00, Jan 24 '24 at 06:41
@azif006 So are we 'squaring' the 'd' in the numerator? And why aren't we 'squaring' the d in the denominator? — Krrish Gupta, Jan 24 '24 at 06:47
The reason we write the square of $dx$ as $dx^2$ instead of $d^2x^2$ is the same as the reason we write $\sin^2$ instead of $s^2i^2n^2$. It's because $dx$ is not the product of $d$ and $x$, it's a variable with a two-letter name. — user14111, Jan 24 '24 at 07:04
The numerator $d^2f$ represents an operation, like log, square root, etc. The square says take the operation twice. If you go further in Calculus you'll encounter $\frac{d^2f}{dx dy}$, which means take the derivative relative to y first, then the derivative relative x. For first semester Calculus courses your denominator, could be written $dxdx$ — nickalh, Jan 24 '24 at 07:13
Please search on the website to see if your question has been answered before. I bet this one has. Duplicate questions are likely to be closed. — nickalh, Jan 24 '24 at 07:14
@KrrishGupta because $\frac{dy^2}{dx^2}$ means $\frac{dy}{dx}\frac{dy}{dx}$, which is very different from $\frac{d}{dx}\frac{d}{dx} y$. — Yi Jiang, Jan 24 '24 at 07:26
Okay one more question @nickalh: Like in exponentiation, we can raise any number to any real power, can we differentiate any number of times? For example, does $\frac{d^{1.5}y}{dx^{1.5}}$ have any meaning? — Krrish Gupta, Jan 24 '24 at 08:41
I have a B.S. in Math and have never fractional powers for derivatives. They may exist somewhere but it's not in common usage at the undergraduate level. — nickalh, Jan 24 '24 at 08:43
There are fractional derivatives, you can checkout the wikipedia page. — Trebor, Jan 24 '24 at 08:59
https://math.stackexchange.com/questions/475016/leibniz-notation-for-high-order-derivatives/4831211#4831211 — zkutch, Jan 24 '24 at 11:33

score 2 · Answer 1 · answered Jan 24 '24 at 07:20

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.

score 1 · Answer 2 · answered Jan 24 '24 at 08:58

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.

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

2 Answers2