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) = (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 - 2 + sqrt(x))`   ...[By (1)]

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

∴ `"dy"/"dx" = (1)/(("dx"/"dy")`
= `-(1)/(4sqrt(x)sqrt(2 - sqrt(x)`.