Question

Differentiate the following w.r.t. x: 

sin (tan^–1 e^–x)

Answer 1

Let, sin `(tan^-1 e^(- x))`

Differentiating both sides with respect to x,

`dy/dx = d/dx sin(tan^-1 e^(-x))`

`= cos (tan^-1 e^-x) d/dx  tan^-1 e^-x`

`= cos (tan^-1 e^-x) 1/(1 + (e^-x)^2) d/dx  (e^-x)`

`= cos (tan^-1 e^-x) 1/(1 + e^(-2x)) (e^-x) d/dx (-x)`

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

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