Derivation/Inverse function/False proof/Exercise

Let

$f:ℝ⟶ℝ$

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

We have

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

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

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

Hence,

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

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