Find the derivative of f(tan x) w.r.t. g (sec x) at `x=(pi)/(4),` where f'(1)=2 and `g'(sqrt(2))=4.`

Options

A. `(1)/(sqrt(2))`
B. `sqrt(2)`
C. 1
D. 0

Answer

A

Explanation

Let `u=f(tanx)" and "v=g(secx)`. Then,
`(du)/(dv)=((du)/(dx))/((dv)/(dx))=(f'(tanx)sec^(2)x)/(g'(secx)secxtanx)`
`implies" "(du)/(dv)=(f'(tanx))/(g'(secx))cosecx`
`implies" "((du)/(dv))_(x=(pi)/(4))=sqrt(2)(f')/(g')((1))/((sqrt(2)))=(2sqrt(2))/(4)=(1)/(sqrt(2))`