Question

Suppose f(π/3) = 4 and f '(π/3) = −5, and let g(x) = f(x) sin(x) and h(x) = cos(x)/f(x). Find the following.g'(π/3) h'(π/3)

Answer 1

Use product rule and quotient rule, respectively. g'(x)=f(x)cos(x)+f'(x)sin(x). Plugging in Pi/3, we get 4(1/2)+(-5)(sqrt(3)/2).
For h'(x), we use quotient rule to find that h'(x)= (f(x)*(-sin(x))-cos(x)*f'(x))/(f(x))^2.
So h'(pi/3)= (4(-sqrt(3)/2)-(1/2)(-5))/16.