Let g(x)=cos(pi x), and f(x)=g(g(x)). Define a= f(1/3). Fine the value of a. Find f'(1/3)

Question

Let g(x)=cos(pi x), and f(x)=g(g(x)). Define a= f(1/3).

Fine the value of a.

Find f'(1/3).

Answer 1

g(x) = cos(πx), f(x) = g(g(x)) = cos(πcos(πx))

a = f(1/3) = cos(πcos(π/3)) = cos(π(1/2)) = 0

f ' (x) = g '(g(x))g '(x) by the chain rule.

= -sin(πcos(πx))(-πsin(πx))

f '(1/3) = -sin(π/2)(-πsin(π/3)) = π(1)(√3)/2) = (√3)π/2