Question

Differentiate the following w.r.t. x :

`sin^(−1) ((1 − x^3)/(1 + x^3))`

Answer 1

Let `y = sin^(−1)  ((1 − x^3)/(1 + x^3))`

`y = sin^(−1)[(1 − (x^(3/2))^2)/(1 + (x^(3/2))^2)]`

Put `x^(3/2) = tan θ. "Then"  θ = tan^(−1)(x^(3/2))`

∴ y = `sin^(−1)((1 − tan^2θ)/(1 + tan^2θ))`

∴ y = sin⁻¹(cos 2θ)

∴ y = `[sin(π/2 − 2θ)]`

∴ y = `π/(2) − 2θ`

∴ y = `π/(2) − 2tan^(−1)(x^(3/2))`

Differentiating w.r.t. x, we get

`dy/dx = d/dx [π/2 − 2tan^(−1) (x^(3/2))]`

`dy/dx = d/dx (π/2) − 2d/dx [tan^(−1) (x^(3/2))]`

`dy/dx = 0 − 2 × (1)/(1 + (x^(3/2))^2). d/dx (x^(3/2))`

`dy/dx = − (2)/(1 + x^3) × (3)/(2)x^(1/2)`

`dy/dx = −(3sqrt(x))/(1 + x^3)`