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$(f^{-1})'(y)=f'(f^{-1}(y))^{-1}$

This is the formula of the derivative of the inverse. As far as I understand, $f^{-1}(y)$ should give back $y$ (as we have the inverse function of $f(x)$ which should be different from the inverse of $f(x)$). Then, what is the difference between $(f^{-1})'(y)$ and $f'(y)^{-1}$? Does the order of the derivative and inverse notation matter? Thank you for the help!

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  • $\begingroup$ Is there a difference between $f^-1 (f(x))$ and $f^-1(x)$ ? Thank you $\endgroup$ Commented Dec 30, 2022 at 17:32
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    $\begingroup$ I edited your question so that the entire $^{-1}$ appears as a superscript. You need to put curly braces {} around the superscript to do this: ^{-1} instead of ^-1. $\endgroup$ Commented Dec 30, 2022 at 17:34
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    $\begingroup$ If $F$ is a function, $F’(x)$ is the derivative of $F$ at $x$. So, $(f^{-1})’(y)$ is the derivative of $f^{-1}$ at $y$; while $f’(f^{-1}(y))$ is the derivative of $f$ at $f^{-1}(y)$; and $f’(f^{-1}(y))^{-1}$ is just $$\frac1{f’(f^{-1}(y))}.$$ $\endgroup$
    – azif00
    Commented Dec 30, 2022 at 17:35

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There are two different meanings of the superscript $ ^ { - 1 } $ here: when applied to an expression for a function, it means the inverse function; when applied to an expression for a number, it means the reciprocal. So $ f ^ { - 1 } ( y ) $ means to apply the inverse function of $ f $ to the number $ y $, while $ f ( y ) ^ { - 1 } $ means the reciprocal of the number $ f ( y ) $. If you throw derivatives in there, then $ ( f ^ { - 1 } ) ' ( y ) $ means that you take the inverse function of $ f $, differentiate that inverse function, and then apply that derivative to $ y $; while $ f ' ( y ) ^ { - 1 } $ means that you differentiate the original function $ f $, then apply that derivative to $ y $, then take the reciprocal of the result.

Since the reciprocal can also be written using a division symbol, I think that it would be less confusing to write $ 1 / f ( y ) $ or $ \frac 1 { f ( y ) } $ instead of $ f ( y ) ^ { - 1 } $. Then the rule for differentiating an inverse function becomes $$ ( f ^ { - 1 } ) ' ( y ) = \frac 1 { f ' ( f ^ { - 1 } ( y ) ) } $$ which might make more sense. If $ y = f ( x ) $, so that $ f ^ { - 1 } ( y ) = x $, then you can also write this rule as $$ ( f ^ { - 1 } ) ' ( y ) = \frac 1 { f ' ( x ) } $$ if you prefer.

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