Question

Find `bb(dy/dx)` in the following:

`y = tan^(-1) ((3x -x^3)/(1 - 3x^2)), - 1/sqrt3 < x < 1/sqrt3`

Answer 1

y = `tan^-1 ((3x - x^3)/(1 - 3x^2))`

Putting x = tan θ,

∴ y = `tan^-1 ((3 tan theta - tan^3 theta)/(1 - 3 tan^2 theta))`

= tan⁻¹ (tan 3 θ)

= 3 θ 

= 3 tan⁻¹ x  ...[because θ = tan⁻¹ x]

On differentiating with respect to x,

`dy/dx = 3 d/dx tan^-1 x`

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

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