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 = `root(3)(x - 2)`

Answer 1

y = `root(3)(x - 2)`                            ...(1)
We have to find the inverse function of y = f(x), i.e x in terms of y.
From (1),
y³ = x – 2
∴ x = y³ + 2
∴ x = f^–1(y) = y³ + 2
∴ `"dx"/"dy" = "d"/"dy"(y^3 + 2)`
= 3y² + 0 = 3y²
= `3(root(3)((x - 2)))^2`                    ...[By (1)]
= `3(x - 2)^(2/3)`
= `3.(root(3)((x - 2)^2))`
∴ `"dy"/"dx" = (1)/(("dx"/"dy")`
= `(1)/(3root(3)((x - 2)^2)), x > 2`.