The functions f and g are differentiable and f(g(x))=tanx for all x. If f(pi/4)=0, g(pi/4)=pi/3 and f'(pi/3)=1/2, what is the value of g'(pi/4)?

Question

Possible answers include:

 

A. 0

B. 1

C. 2

D. 4

Answer 1

If F(x) = f(g(x)) then F′(x) = f′(g(x))g′(x)  (chain rule)

 

f(g(x))=tan(x)  (take the derivative of both side using the chain rule)

f′(g(x))g′(x)=sec²(x)  let x=π/4  -->

f′(g(π/4))g′(π/4)=2      but  g(π/4)=π/3  and  f'(π/3)= 1/2    so

f'(π/3)g′(π/4)=2  -->  (1/2)g′(π/4)=2  so  g′(π/4)=4