Let
-
be a bijective differentiable function with
for all
,
and the inverse function
. What is wrong in the following "Proof“ for the derivative of the inverse function?
We have
-
![{\displaystyle {}{\left(f\circ f^{-1}\right)}(y)=y\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c043c9ba282d4611d1e486b003e29cd1272ab68)
Using the chain rule, we get by differentiating on both sides the equality
-
![{\displaystyle {}f'{\left(f^{-1}(y)\right)}{\left(f^{-1}\right)}'(y)=1\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb8c354b7a1a9a3f265b01c63611f0b0d8230d4)
Hence,
-
![{\displaystyle {}{\left(f^{-1}\right)}'(y)={\frac {1}{f'{\left(f^{-1}(y)\right)}}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8846676f59970adc0d22d444cdec1bc14ce84efb)