Final answer:
To find the derivative of f(x) = sin(g(x)), use the chain rule to differentiate. Given g(-4) = π/2 and g'(-4) = -2, find f'(-4). The derivative of f(x) is f'(x) = cos(g(x)) * g'(x), so substitute the values to find f'(-4) = 0.
Step-by-step explanation:
To find the derivative of f(x) = sin(g(x)), we need to use the chain rule. The chain rule states that if we have a composition of two differentiable functions, f(g(x)), then the derivative is given by f'(g(x)) * g'(x). Applying this to our function, we have f'(x) = cos(g(x)) * g'(x).
Given that g(-4) = π/2 and g'(-4) = -2, we can plug these values into our derivative formula: f'(-4) = cos(g(-4)) * g'(-4). Since g(-4) = π/2, cos(g(-4)) = cos(π/2) = 0. And g'(-4) = -2. Therefore, f'(-4) = 0 * -2 = 0.