Derivation/Inverse function/False proof/Exercise

Let

$f\colon \mathbb {R} \longrightarrow \mathbb {R}$

be a bijective differentiable function with ${}f'(x)\neq 0$ for all ${}x\in \mathbb {R}$, and the inverse function ${}f^{-1}$. What is wrong in the following "Proof“ for the derivative of the inverse function?

We have

${}{\left(f\circ f^{-1}\right)}(y)=y\,.$

Using the chain rule, we get by differentiating on both sides the equality

${}f'{\left(f^{-1}(y)\right)}{\left(f^{-1}\right)}'(y)=1\,.$

Hence,

${}{\left(f^{-1}\right)}'(y)={\frac {1}{f'{\left(f^{-1}(y)\right)}}}\,.$

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