Of y=f(x) is a differentiable function of x such that inverse function x=1/f(y) exists then prove that x is a differentiable function of y and dx/dy=1/dy/dx, when dy/dx is not equal to zero.

We will let inverse function exist and callcit f-1

Then we want to determine if y= f-1(x) is differentiable or not

But f(y)=x

f'(y)*dy/dx=1

dy/dx=1/f'(y), but as it is given that f is differentiable that means f' exists for all y, and hence the inverse function is differentiable with its differential= 1/f'(y) as long as the denominator is not zero.