Question

If y = f(x) is differentiable function of x such that inverse function x = f^–1 (y) exists then prove that x is a differentiable function of y and dx/dy = 1/(dy/dx) where  dy/ dx ≠0

Answer 1

Let \(\delta y\) be the increment in y corresponding to an increment dx in x.
as \(\delta x\) → 0, \(\delta\)у → 0

Now y is a differentiable function of x.

Taking limits on both sides as \(\delta x \rightarrow 0 , \) we ≥ t

Since limit in R.H.S. exists

limit in L.H.S. also exists and we have,