Given the function f(x)=2(-9x^(2)+x+6)^(4), 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^4 where u is the inner function, and the inner function is -9x^2 + x + 6. 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.Differentiate Outer Function: Differentiate the outer function with respect to the inner function u. The derivative of u^4 with respect to u is 4u^3.Differentiate Inner Function: Differentiate the inner function -9x^2 + x + 6 with respect to x. The derivative of -9x^2 is -18x, the derivative of x is 1, and the derivative of 6 is 0. So, the derivative of the inner function is -18x + 1.Apply Chain Rule: Apply the chain rule to find f'(x). f'(x) = 2 * 4(-9x^2 + x + 6)^3 * (-18x + 1)Simplify Expression: Simplify the expression for f'(x). f'(x) = 8(-9x^2 + x + 6)^3 * (-18x + 1)

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Given the function
f(x)=2(-9x^(2)+x+6)^(4), find
f^(')(x) in any form.
Answer:
f^(')(x)=

Given the function $f(x)=2\left(-9 x^{2}+x+6\right)^{4}$ , 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

Full solution

Q. Given the function

f(x)=2\left(-9 x^{2}+x+6\right)^{4}

, 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 $u^4$ where $u$ is the inner function, and the inner function is $-9x^2 + x + 6$ . 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.
Differentiate Outer Function: Differentiate the outer function with respect to the inner function $u$ . The derivative of $u^4$ with respect to $u$ is $4u^3$ .
Differentiate Inner Function: Differentiate the inner function $-9x^2 + x + 6$ with respect to $x$ . The derivative of $-9x^2$ is $-18x$ , the derivative of $x$ is $1$ , and the derivative of $6$ is $0$ . So, the derivative of the inner function is $-18x + 1$ .
Apply Chain Rule: Apply the chain rule to find $f'(x)$ . $\newline$ $f'(x) = 2 \cdot 4(-9x^2 + x + 6)^3 \cdot (-18x + 1)$
Simplify Expression: Simplify the expression for $f'(x)$ . $f'(x) = 8(-9x^2 + x + 6)^3 \cdot (-18x + 1)$