Question

Find `("d"y)/("d"x)`, if y = `tan^-1 ((3x - x^3)/(1 - 3x^2)), -1/sqrt(3) < x < 1/sqrt(3)`

Answer 1

Put x = tan θ

Where `(-pi)/6 < θ < pi/6`

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

= `tan^-1 (tan 3theta)`

= 3_θ  ...`(because (-pi)/2 < 3theta < pi/2)`

= 3tan^–1x

Hence, `("d"y)/("d"x) = 3/(1 + x^2)`