Jika f(pi/3)=1, f'(pi/3)=-2g(x)=f(x)sin x, dan h(x)=(cos

g'(π/3) = (1 - 2√3)/2, h'(π/3) = (√3 - 2)/2.

Pembahasan

Untuk menentukan nilai g'(π/3) dan h'(π/3):

Diberikan:
f(π/3) = 1
f'(π/3) = -2
g(x) = f(x) sin x
h(x) = (cos x) / f(x)

a. Menentukan g'(π/3):
Kita gunakan aturan perkalian untuk turunan: g'(x) = f'(x) sin x + f(x) cos x.
Substitusikan x = π/3:
g'(π/3) = f'(π/3) sin(π/3) + f(π/3) cos(π/3)
g'(π/3) = (-2) * (√3/2) + (1) * (1/2)
g'(π/3) = -√3 + 1/2
Jadi, g'(π/3) = (1 - 2√3) / 2.

b. Menentukan h'(π/3):
Kita gunakan aturan pembagian untuk turunan: h'(x) = [f'(x) cos x - f(x) (-sin x)] / [f(x)]².
Substitusikan x = π/3:
h'(π/3) = [f'(π/3) cos(π/3) - f(π/3) (-sin(π/3))] / [f(π/3)]²
h'(π/3) = [(-2) * (1/2) - (1) * (-√3/2)] / [1]²
h'(π/3) = [-1 + √3/2] / 1
h'(π/3) = -1 + √3/2
Jadi, h'(π/3) = (√3 - 2) / 2.