Find the derivative of `f(tanx)` w.r.t. `g(secx)` at `x=pi/4` , where `f^(prime)(1)=2` and `g^(prime)(sqrt(2))=4` .

Options

A. `sqrt(2)`
B. `(1)/(sqrt(2))`
C. 1
D. none of these

Answer

B

Explanation

Let u =F (tan x) and v= g ( sec x)
`implies (du)/(dx)=f'(tanx) sec^(2)x`
`and (dv)/(dx)=g' (sec x0 sec x tan x`
`implies(du)/(dv)=(du)/(dx)//(dv)/(dx)=(f' (tan x)sec^(2)x)/(g'(secx)sec x tan x)`
`[(du)/(dv)]_(x=(pi)/(4))=(f'(tan""(pi)/(4)))/(g'(sec""(pi)/(4))sin""(pi)/(4))=(f'(1)sqrt(2))/(g'(sqrt(2)))`
`=(2sqrt(2))/(4)=(1)/(sqrt(2))`