Question

Differentiate the following w.r.t. x : `tan^-1((sqrt(x)(3 - x))/(1 - 3x))`

Answer 1

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

= `tan^-1[(3sqrt(x) - xsqrt(x))/(1 - 3x)]`

Put `sqrt(x) = tanθ`. Then θ = `tan^-1(sqrt(x))`

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

= tan^–1 (tan3θ)

= 3θ

= `3tan^-1(sqrt(x))`

∴ `"dy"/"dx" = 3"d"/"dx"[tan^-1(sqrt(x))]`

= `3 xx (1)/(1 + (sqrt(x))^2)."d"/"dx"(sqrt(x))`

= `(3)/(1 + x) xx (1)/(2sqrt(x))`

= `(3)/(2sqrt(x)(1 + x)`.