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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 $f$?

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    $\begingroup$ I prefer $f'(x)^2$. $\endgroup$
    – GEdgar
    May 19 '20 at 1:16
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    $\begingroup$ $\tan^{-1} (x)$ does not mean the same thing as $\tan(x)^{-1}$ $\endgroup$
    – Sandejo
    May 19 '20 at 1:17
  • $\begingroup$ @Sandejo Edited. $\endgroup$ May 19 '20 at 1:18
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    $\begingroup$ I prefer $$\left( f'(x) \right)^2$$ $\endgroup$
    – N. S.
    May 19 '20 at 1:32
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    $\begingroup$ 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. $\cos^{2(2)}(x) = -2\cos(2x)$. $\endgroup$ May 19 '20 at 12:22
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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 $f$.

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    $\begingroup$ +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. $\endgroup$ May 19 '20 at 4:56
  • $\begingroup$ @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. $\endgroup$ May 20 '20 at 8:32

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