Question

Differentiate the following w.r.t. x :

`cos^-1[(3cos(e^x) + 2sin(e^x))/sqrt(13)]`

Answer 1

y = `cos^-1((3cos(e^x) + 2sin(e^x))/sqrt(13))`

y  = `cos^-1(cos(e^x).(3)/sqrt(13) + sin(e^x)2/sqrt(13))`

Put, `(3)/sqrt(13) = cos α, (2)/sqrt(13) = sin α`

Also,

sin²α + cos²α = `(9)/(13) + (4)/(13)` = 1

And,

tan α = `(2)/(3)       ∴ α = "tan"^-1(2/3)`

y = cos^–1(cos e^x. cos α + sin e^x. sin α)

y = cos^–1[cos(e^x – α)]       ...[∵ cos^–1x.(cos x) = x]

y = e^x – α

y = `e^x  –  tan^-1(2/3)`

Differentiating w.r.t. x, we get

`dy/dx = d/dx [e^x - tan^-1(2/3)]`

`dy/dx` = e^x – 0

`dy/dx` = e^x