Question

Differentiate the following w.r.t. x:

`sin^-1  ((1 - 25x^2)/(1 + 25x^2))`

Answer 1

Let y = `sin^-1  ((1 - 25x^2)/(1 + 25x^2))`

= `sin^-1[(1 - (5x)^2)/(1 + (5x)^2)]`
Put 5x = tanθ.
Then θ = tan^–1(5x)

∴ y = `sin^-1((1 - tan^2θ)/(1 + tan^2θ))`
= sin^–1(cos2θ)

= `sin^-1[sin(pi/2 - 2θ)]`

= `pi/(2) - 2θ`

= `pi/(2) - 2tan^-1(5x)`
Differentiating w.r.t. x, we get
∴ `(dy)/(dx) = (d)/(dx)[pi/2 - 2tan^-1 (5x)]`

= `d/(dx)(pi/2) - 2 d/(dx)[tan^-1(5x)]`

= `0 - 2 xx (1)/(1 + (5x)^2).d/(dx)(5x)`

= `(-2)/(1 + 25x^2) xx 5`

= `(-10)/(1 + 25x^2)`.