Given the function f(x)=(-2x^(2)+6x)^(6), find f^(')(x) in any form. Answer: f^(')(x)=

Identify Functions: To find the derivative of the function f(x) = (-2x^2 + 6x)^6, we will use the chain rule. 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.Find Derivatives: First, let's identify the outer function and the inner function. The outer function is g(u) = u^6 and the inner function is u(x) = -2x^2 + 6x. We will need to find the derivative of both functions.Apply Chain Rule: The derivative of the outer function g(u) = u^6 with respect to u is g'(u) = 6u^5.Substitute Derivatives: The derivative of the inner function u(x) = -2x^2 + 6x with respect to x is u'(x) = -4x + 6.Simplify Expression: Now, we apply the chain rule: f'(x) = g'(u(x)) * u'(x). Substituting the derivatives we found, we get f'(x) = 6(-2x^2 + 6x)^5 * (-4x + 6).Final Derivative Form: Simplify the expression to get the final form of the derivative: f'(x) = 6(-2x^2 + 6x)^5 * (-4x + 6). This is the derivative of the function f(x) in any form.

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Given the function
f(x)=(-2x^(2)+6x)^(6), find
f^(')(x) in any form.
Answer:
f^(')(x)=

Given the function $f(x)=\left(-2 x^{2}+6 x\right)^{6}$ , 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)=\left(-2 x^{2}+6 x\right)^{6}

, find

f^{\prime}(x)

in any form.

\newline

Answer:

f^{\prime}(x)=

Identify Functions: To find the derivative of the function $f(x) = (-2x^2 + 6x)^6$ , we will use the chain rule. 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.
Find Derivatives: First, let's identify the outer function and the inner function. The outer function is $g(u) = u^6$ and the inner function is $u(x) = -2x^2 + 6x$ . We will need to find the derivative of both functions.
Apply Chain Rule: The derivative of the outer function $g(u) = u^6$ with respect to $u$ is $g'(u) = 6u^5$ .
Substitute Derivatives: The derivative of the inner function $u(x) = -2x^2 + 6x$ with respect to $x$ is $u'(x) = -4x + 6$ .
Simplify Expression: Now, we apply the chain rule: $f'(x) = g'(u(x)) \cdot u'(x)$ . Substituting the derivatives we found, we get $f'(x) = 6(-2x^2 + 6x)^5 \cdot (-4x + 6)$ .
Final Derivative Form: Simplify the expression to get the final form of the derivative: $f'(x) = 6(-2x^2 + 6x)^5 * (-4x + 6)$ . This is the derivative of the function $f(x)$ in any form.