Question

Find the derivative of the function y = f(x) using the derivative of the inverse function x = f^–1(y) in the following:

y = `sqrt(x)`

Answer 1

y = `sqrt(x)`  ...(1)

We have to find the inverse function of y = f(x), i.e. x in terms of y.

From (1), we have

y² = x

∴ x = y²

∴ x = f^–1(y) = y²

∴ `(dx)/(dy) = d/(dy)(y^2)` = 2y

= `2sqrt(x)`   ...[By (1)]

∴ `(dy)/(dx) = (1)/((dx/dy)` 

= `(1)/(2sqrt(x)`.