Answer and Step-by-step explanation:
To find the derivative of f(x) at x = -1, we can use the product rule and chain rule of differentiation.
Given that f(x) = x^10 * h(x), we need to differentiate both parts separately and then multiply them together.
Let's start with differentiating x^10:
Using the power rule, the derivative of x^10 is 10x^9.
Next, let's differentiate h(x) with respect to x:
Since we don't have the explicit expression for h(x), we can use the chain rule.
The chain rule states that if we have a function g(x) = h(f(x)), then the derivative of g(x) with respect to x is given by g'(x) = h'(f(x)) * f'(x).
In this case, h(x) is evaluated at x = -1, so h'(-1) represents the derivative of h(x) at x = -1.
Now, we have f(x) = x^10 * h(x), so we can differentiate it using the product rule:
f'(x) = (10x^9 * h(x)) + (x^10 * h'(x))
Substituting x = -1 into f'(x), we get:
f'(-1) = (10 * (-1)^9 * h(-1)) + ((-1)^10 * h'(-1))
Since (-1)^9 = -1 and (-1)^10 = 1, we can simplify the expression:
f'(-1) = -10h(-1) + h'(-1)
Given that h(-1) = 5 and h'(-1) = 8, we can substitute these values into the equation:
f'(-1) = -10(5) + 8
Evaluating the expression, we get:
f'(-1) = -50 + 8
Finally, we simplify to find the value of f'(-1):
f'(-1) = -42
Therefore, the derivative of f(x) at x = -1, f'(-1), is equal to -42.