# Is it common to write $f'^{2}(x)$, particularly when $f$ is a trigonometric function?

It has been said many times that the notation for exponentiated or inverted trig functions, e.g.:

$$\sin^2(x), \tan^{-1}(x), \csc^3(x)$$

is confusing, ugly, and terrible in general, but nevertheless standard. So is it more common to write: $$f'^2(x)\text{ or }f'(x)^2$$ To differentiate and then square a function $$f$$?

• I prefer $f'(x)^2$. May 19 '20 at 1:16
• $\tan^{-1} (x)$ does not mean the same thing as $\tan(x)^{-1}$ May 19 '20 at 1:17
• @Sandejo Edited. May 19 '20 at 1:18
• I prefer $$\left( f'(x) \right)^2$$ May 19 '20 at 1:32
• People also sometimes write $f^{(n)}$ for the $n$th derivative of $f$. You could no doubt abuse this to make some really confusing expressions. $\cos^{2(2)}(x) = -2\cos(2x)$. May 19 '20 at 12:22

In your example, you use superscripts to mean two different things. In $$\sin^2$$ and $$\csc^3$$, the superscripts refer to iterated multiplication (powers), whereas in $$\tan^{-1}$$, the superscript refers to iterated composition (inverse for $$-1$$). While this is not ambiguous for trig functions as there is an established convention of positive superscripts referring to powers and using $$-1$$ for the inverse, this is not the case more generally, where $$f^n$$ could refer to a power of $$f$$ or an iterated function. For that reason, it is clearer to write $$f^\prime(x)^2$$ to indicate the square of the derivative of $$f$$.
• +1. I suspect that the common trig notations like $\sin^2 x$ for $(\sin x)^2$ arose when someone got tired of writing so many brackets. May 19 '20 at 4:56
• @DanielWainfleet: good point. Apart from non-ambiguity simplicity is also desired in a notation. No one really wants to write unnecessary parentheses. When I write $\sin x^2$ I mean $\sin (x^2)$ and not $(\sin x) ^2$. Unfortunately function other than trig don't seem to have similar convention. May 20 '20 at 8:32