Question

Differentiate the following w.r.t. x : `cos^-1((e^x -  e^(-x))/(e^x +  e^(-x)))`

Answer 1

Let y = `cos^-1((e^x -  e^(-x))/(e^x +  e^(-x)))`

= `cos^-1[(e^x - 1/e^x)/(e^x + 1/e^x)]`

= `cos^-1((e^(2x) - 1)/(e^(2x) + 1))`
Put e^x = tanθ.
Then θ = tan^–1(e^x)

∴ y = `cos^-1((tan^2θ - 1)/(tan^2θ + 1))`

= `cos^-1[-((1 - tan^2θ)/(1 + tan^2θ))]`

= cos^–1(– cos2θ)
= cos^–1[cos(π – 2θ)]
= π – 2θ
= π – 2tan^–1(e^x)
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"[pi - 2tan^-1(e^x)]`

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

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

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

= `-(2e^x)/(1 + e^(2x)`