# Notation for derivatives of $\sin x$. Can we use $\sin^\prime x$?

I was teaching Trig Derivatives today to a high school class and someone asked a question I’ve never had before:

Can we use $\sin^\prime x$ to represent the derivative of $\sin x$?

I have never seen that used when studying math and a Google search only showed one use.

However when I googled $\sin^\prime x$ it came back with hits for the derivative of $\sin x$. We do use a similar notation for the inverse and square of $\sin x$, so why not the derivative?

Any thoughts would be appreciated. Thanks.

• Yes, sin'(x) is not an uncommon notation though (sin(x))' and d(sin(x))/dx are more common. – user247327 May 7 '18 at 17:41

The notation you describe has a certain appeal, however it is not in common use. The usual shorthands are the notations I'm sure you're familiar with, such as $\frac{d}{dx}\sin x$, or denoting the function as $y$ or $f(x)$, and then talking about $y'$ or $f'(x)$.

I don't know of a reason that the "prime" notation isn't used in the way you describe, but we've been writing calculus down for so long now, that it basically comes down to inertia. Even if a new notation in basic calculus is great, it would take a massive effort to render it recognizable on any large scale, and most of us have better things to do with our time.

In other words, as long as you use the notation consistently, there's nothing logically wrong with it, but it's by no means a standard, and it's likely to appear funny-looking to the majority of mathematics readers. Whether you want to teach a non-standard notation in your class is entirely up to you. I think such a move can sometimes be warranted, and sometimes just cause confusion down the line.

The only reason it's not in common use is that we know $\sin' = \cos$. For other "named" special functions whose derivatives don't have a closed form, the analogous notation is often used, e.g. $\text{Ai}'(x)$ and $\text{Bi}'(x)$ for the Airy functions.

If I'm reading between the lines correctly, it seems like you're concerned that putting the prime before the parentheses is at odds with how we would write derivatives for other functions, just like how the notation $\sin^2(x)$ is at odds with how we would normally write $f(x)^2$.

But that just isn't true. We normally would write the derivative of $f$ as $f'(x)$. Strictly speaking $(f(x))'$ could even be considered wrong, being the derivative of the constant function $t\to f(x)$ for some fixed $x$.

It's just that because $\sin$ is a named function, the notation $\sin'(x)$ looks vaguely like you might be talking about some new function that just happens to be called $\sin'$.