Given the function f(x)=(-x^(2)-8x-8)^(5), find f^(')(x) in any form. Answer: f^(')(x)=

Identify Functions: Identify the outer function and the inner function for the application of the chain rule. The outer function is u^5 and the inner function is u = -x^2 - 8x - 8.Apply Chain Rule: Apply the chain rule which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Let u = -x^2 - 8x - 8, then f(x) = u^5. The chain rule gives us f'(x) = 5u^4 * (du/dx).Calculate Derivative: Calculate the derivative of the inner function u with respect to x. du/dx = d/dx(-x^2 - 8x - 8) = -2x - 8.Substitute into Formula: Substitute the derivative of the inner function and the inner function itself into the chain rule formula. f'(x) = 5(-x^2 - 8x - 8)^4 * (-2x - 8).Simplify Expression: Simplify the expression if necessary. In this case, the expression is already in a simplified form, so we can state the final answer. f'(x) = 5(-x^2 - 8x - 8)^4 * (-2x - 8).

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Given the function
f(x)=(-x^(2)-8x-8)^(5), find
f^(')(x) in any form.
Answer:
f^(')(x)=

Given the function $f(x)=\left(-x^{2}-8 x-8\right)^{5}$ , find $f^{\prime}(x)$ in any form. $\newline$ Answer: $f^{\prime}(x)=$

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Algebra 2

Find the vertex of the transformed function

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Q. Given the function

f(x)=\left(-x^{2}-8 x-8\right)^{5}

, find

f^{\prime}(x)

in any form.

\newline

Answer:

f^{\prime}(x)=

Identify Functions: Identify the outer function and the inner function for the application of the chain rule. $\newline$ The outer function is $u^5$ and the inner function is $u = -x^2 - 8x - 8$ .
Apply Chain Rule: Apply the chain rule which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. $\newline$ Let $u = -x^2 - 8x - 8$ , then $f(x) = u^5$ . $\newline$ The chain rule gives us $f'(x) = 5u^4 \cdot (\frac{du}{dx})$ .
Calculate Derivative: Calculate the derivative of the inner function $u$ with respect to $x$ . $\frac{du}{dx} = \frac{d}{dx}(-x^2 - 8x - 8) = -2x - 8.$
Substitute into Formula: Substitute the derivative of the inner function and the inner function itself into the chain rule formula. $\newline$ $f'(x) = 5(-x^2 - 8x - 8)^4 \cdot (-2x - 8)$ .
Simplify Expression: Simplify the expression if necessary. $\newline$ In this case, the expression is already in a simplified form, so we can state the final answer. $\newline$ $f'(x) = 5(-x^2 - 8x - 8)^4 * (-2x - 8)$ .