Given the function f(x)=(8x-6)^(6), find f^(')(x) in any form. Answer: f^(')(x)=

Identify Function: Identify the function to differentiate. We are given the function f(x) = (8x - 6)^6. We need to find its derivative, denoted as f'(x).Apply Chain Rule: Apply the chain rule for differentiation. The chain rule 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. In this case, the outer function is u^6 and the inner function is u = 8x - 6.Differentiate Outer Function: Differentiate the outer function with respect to the inner function. The derivative of u^6 with respect to u is 6u^5. We will substitute u back with 8x - 6 later.Differentiate Inner Function: Differentiate the inner function with respect to x. The derivative of 8x - 6 with respect to x is 8, since the derivative of a constant is 0 and the derivative of 8x is 8.Combine Using Chain Rule: Combine the results using the chain rule. Now we multiply the derivative of the outer function by the derivative of the inner function: f'(x) = 6(8x - 6)^5 * 8.Simplify Expression: Simplify the expression. We can simplify the expression by multiplying 6 and 8 together: f'(x) = 48(8x - 6)^5.

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Given the function
f(x)=(8x-6)^(6), find
f^(')(x) in any form.
Answer:
f^(')(x)=

Given the function $f(x)=(8 x-6)^{6}$ , 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)=(8 x-6)^{6}

, find

f^{\prime}(x)

in any form.

\newline

Answer:

f^{\prime}(x)=

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $f(x) = (8x - 6)^6$ . We need to find its derivative, denoted as $f'(x)$ .
Apply Chain Rule: Apply the chain rule for differentiation. $\newline$ The chain rule 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. In this case, the outer function is $u^6$ and the inner function is $u = 8x - 6$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function. $\newline$ The derivative of $u^6$ with respect to $u$ is $6u^5$ . We will substitute $u$ back with $8x - 6$ later.
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The derivative of $8x - 6$ with respect to $x$ is $8$ , since the derivative of a constant is $0$ and the derivative of $8x$ is $8$ .
Combine Using Chain Rule: Combine the results using the chain rule. $\newline$ Now we multiply the derivative of the outer function by the derivative of the inner function: $f'(x) = 6(8x - 6)^5 \times 8$ .
Simplify Expression: Simplify the expression. $\newline$ We can simplify the expression by multiplying $6$ and $8$ together: $f'(x) = 48(8x - 6)^5$ .