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(2 - sqrt(x)`

Answer 1

y = `sqrt(2 - sqrt(x)`    ...(1)

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

From (1),

y² =  `2 - sqrt(x)`

∴ `sqrt(x) = 2 - y^2`

∴ x = (2 – y²)²

∴ x = f⁻¹(y)

x = (2 – y²)²

∴ `(dx)/(dy) = (d)/(dy)(2 - y^2)^2`

= `2(2 - y^2)*(d)/(dy)(2 - y^2)`

= 2(2 – y²).(0 – 2y)

= –4y(2 – y²)

= `-4sqrt(2 - sqrt(x))(2 - (sqrt(2 - sqrt(x)))^2)`   ...[By (1)]

= `-4sqrt(2 - sqrt(x))(2 - 2 + sqrt(x))`

= `-4sqrt(2 - sqrt(x))*sqrt(x)`

= `-4sqrt(x)sqrt(2 - sqrt(x)`

∴ `(dy)/(dx) = (1)/((dx/dy)`

= `-(1)/(4sqrt(x)sqrt(2 - sqrt(x)`.