Final answer:
To evaluate f'(-1) for the function f(x) = x^10h(x) given h(-1) = 3 and h'(-1) = 6, we use the product and chain rule of differentiation. Applying these rules, we find that f'(-1) = -24.
Step-by-step explanation:
To evaluate f'(-1) for the function f(x) = x^10h(x) given that h(-1) = 3 and h'(-1) = 6, we need to apply the product and chain rule of differentiation.
Let's first differentiate f(x) = x^10h(x). Applying the product rule, the derivative of f(x) is f'(x) = 10x^9*h(x) + x^10*h'(x).
Next, we substitute x = -1 into f'(x) to find f'(-1). Given that h(-1) = 3 and h'(-1) = 6, we have f'(-1) = 10*(-1)^9*3 + (-1)^10*6.
Simplifying this expression, we get f'(-1) = -30 + 6 = -24.