# Finding the derivative of an inverse function from first principles

I have tried using the definition of derivative by

$$\lim_{h \to 0} \dfrac{f^{-1}\left(x + h\right) - f^{-1}\left(x\right)}{h}$$

but that is not correct. (it was marked wrong).

What did I do wrong?

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Why do you think that is not correct? I'd say it is, assuming the limit exists. –  DonAntonio Nov 7 '12 at 23:12
Probably, they wanted to express $(f^{-1})'$ using $f'$ and $f$.. –  Berci Nov 7 '12 at 23:34
@berci I think you are correct. –  yiyi Nov 8 '12 at 0:50

## 1 Answer

This is correct so far, but you should go on, somehow introducing the definition of $f'$.

Briefly, it goes like $t:=f^{-1}(x+h)-f^{-1}(x)$, we need that $t\to 0$ as $h\to 0$, and then consider $y:=f^{-1}(x)$ and $$h = (x+h)-x = f(y+t) -f(y)$$

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thanks, I will try to work that path out. –  yiyi Nov 8 '12 at 0:51