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

Identify Functions: We are given the function f(x)=-2(-8x-5)^(5) and we need to find its derivative, which is denoted by f^(')(x). To find the derivative, we will use 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.Find Outer Function Derivative: First, let's identify the outer function and the inner function. The outer function is u^5 where u is the inner function, and the inner function is -8x-5.Find Inner Function Derivative: Now, we will find the derivative of the outer function with respect to u, which is 5u^(5-1) = 5u^4.Apply Chain Rule: Next, we will find the derivative of the inner function with respect to x, which is the derivative of -8x-5. The derivative of -8x is -8, and the derivative of a constant (-5) is 0. So, the derivative of the inner function is -8.Calculate Final Derivative: Now, we apply the chain rule. The derivative of f(x) with respect to x is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This gives us f^(')(x) = 5*(-8x-5)^4 * (-8).Finally, we simplify the expression by multiplying the constants 5 and -8, which gives us f^(')(x) = -40*(-8x-5)^4.

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

Given the function $f(x)=-2(-8 x-5)^{5}$ , 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)=-2(-8 x-5)^{5}

, find

f^{\prime}(x)

in any form.

\newline

Answer:

f^{\prime}(x)=

Identify Functions: We are given the function $f(x)=-2(-8x-5)^{5}$ and we need to find its derivative, which is denoted by $f^{\prime}(x)$ . To find the derivative, we will use 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.
Find Outer Function Derivative: First, let's identify the outer function and the inner function. The outer function is $u^5$ where $u$ is the inner function, and the inner function is $-8x-5$ .
Find Inner Function Derivative: Now, we will find the derivative of the outer function with respect to $u$ , which is $5u^{5-1} = 5u^4$ .
Apply Chain Rule: Next, we will find the derivative of the inner function with respect to $x$ , which is the derivative of $-8x-5$ . The derivative of $-8x$ is $-8$ , and the derivative of a constant $(-5)$ is $0$ . So, the derivative of the inner function is $-8$ .
Calculate Final Derivative: Now, we apply the chain rule. The derivative of $f(x)$ with respect to $x$ is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. $\newline$ This gives us $f'(x) = 5(-8x-5)^4 \times (-8)$ .
Calculate Final Derivative: Now, we apply the chain rule. The derivative of $f(x)$ with respect to $x$ is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. $\newline$ This gives us $f'(x) = 5(-8x-5)^4 \times (-8)$ .Finally, we simplify the expression by multiplying the constants $5$ and $-8$ , which gives us $f'(x) = -40(-8x-5)^4$ .