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.

A) g'(π/3)
B) h'(π/3)

Answer 1

To find g'(π/3), use the product rule and the given information about f(x) and sin(x). To find h'(π/3), use the quotient rule and the given information about f(x) and cos(x).

Step-by-step explanation:

To find g'(π/3), we need to use the product rule and the given information about f(x) and sin(x). Let's start by finding f'(x) using the given information:
f'(x) = -5
Now, we can use the product rule to find g'(π/3):
g'(x) = f'(x)sin(x) + f(x)cos(x)
Plugging in the values:
g'(π/3) = -5sin(π/3) + 4cos(π/3)

To find h'(π/3), we need to use the quotient rule and the given information about f(x) and cos(x). Let's write down the formula for h(x) and its derivative:
h(x) = (cos(x))/f(x)
h'(x) = (f(x)(-sin(x))-cos(x)f'(x))/(f(x))^2
Plugging in the values:
h'(π/3) = ((4)(-sin(π/3))-(cos(π/3))(-5))/(4)^2