# What is the difference between these types of notation symbols in regards to the derivative of inverse functions? Does order matter?

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

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!

• Is there a difference between $f^-1 (f(x))$ and $f^-1(x)$ ? Thank you Dec 30, 2022 at 17:32
• 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. Dec 30, 2022 at 17:34
• 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 $$\frac1{f’(f^{-1}(y))}.$$ Dec 30, 2022 at 17:35

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.