I want to prove the following result: $$ {f^{-1}}'(x)=\frac{1}{f'({f^{-1}}(x))}$$ Is simple application of chain rule a valid proof of it?

Here is an alternative: You can use $$ f^{-1}(x)=\int\frac{1}{f'(f^{-1}(x))}\,dx + c. \tag{1} $$ from "Inverse functions and differentiation".

$\begingroup$ Yes .. if you have two identical functions they have the same derivation. I used this fact twice (for the first implication and for the second implication also) $\endgroup$ – user127.0.0.1

2021年2月17日 · Why is the inverse of the derivative of f not the actual derivative of the inverse of f? 0 An example of a function that is analytic only on the upper-half plane

2022年11月6日 · The inverse function is just a function like any other. So we can differentiate it. In fact we can differentiate g(x) = x - 1 or sqrt(x) without knowing it is an inverse to something. But suppose we know the function is an inverse to something. And we know how to differentiate that something. Is there an easy way to find the derivative of the ...

Existence of a function whose derivative of inverse equals the inverse of the derivative. 4.

2019年1月10日 · I'm a bit confused about how the inverse trig functions are differentiated. From a website: We have the following relationship between the inverse sine function and the sine function. $$\sin \left( {{{\sin }^{ - 1}}x} \right) = x\hspace{0.5in}{\sin ^{ - 1}}\left( {\sin x} \right) = x$$ In other words they are inverses of each other.

2019年11月25日 · How would I calculate the derivative of an inverse function using Leibniz notation, and how does that connect to the process of computing an inverse function's derivative using "prime" notation? calculus

2019年11月25日 · We have a system of three equations for the three second derivatives. Since the equations have become unwieldy, we will switch to the subscript notation for partial derivatives. Use the third equation to substitute for the mixed derivative in the other two equations and substitute in the first derivatives from earlier:

2019年10月8日 · derivative of inverse function problems with proof. 1. Question about the proof for the derivative of the ...