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When asked this specific question

Let $f$ be the function defined on $]\infty,3[$ by $f(x)=x-1+\sqrt{3-x}$.

Find the derivative of the inverse function. (asked before asking to explicit the inverse)

I couldn't find the expression and asked myself : I'm aware that one can find the derivative of inverses of some functions (e.g. trigonometric inverse functions) using formula : $(f^{-1})'(y) = \dfrac{1}{f'(f^{-1}(y))}$ when appliable.

Is this possible for all functions.

My answer would be : it is possible only for functions whose derivatives are expressible in terms of the functions themselves like the tangent function for instance as we have $\tan' = 1+\tan²$.

Would you please confirm and clarify. Thanks.

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    $\begingroup$ the formula for the derivative of the inverse required: a well-defined domain, the expression of the derivative of the original function and an expression for the inverse as well $\endgroup$
    – cards
    Commented Nov 15, 2021 at 14:34

1 Answer 1

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The formula $$(f^{-1})'(y)=\frac{1}{f'[f^{-1}(y)]}$$ is valid as long as both $f^{-1}$ and $f$ are differentiable. The formula is a direct consequence of the chain rule. You start with the equation $$f[f^{-1}(y)]=y,$$ and you differentiate with respect to $y$ both sides. You use the chain rule, and after rearranging, you get the formula you provided. So in fact, it does not work only with trigonometric functions: it works with any invertible differentiable functions.

Of course, even if the formula is valid, that does not mean it is useful. If you do not have an explicit or implicit formula for $f^{-1}$, then it may not be helpful to use that formula in problem-solving. Luckily, in this particular case, finding $f^{-1}$ on some specific interval is fairly easy.

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