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Evaluate f'(-1) for the function f(x) = x^10h(x) given that h(-1) = 3 and h'(-1) = 6.

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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.

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