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Given that
f(x)=x^10h(x)
h(−1)=5
h′(−1)=8,
calculate f′(−1)

User Nam
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1 Answer

5 votes

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.

User Jevgenij Evll
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