Given the function f(x)=-5(5x^(2)+9x)^(6), find f^(')(x) in any form. Answer: f^(')(x)=

Identify Functions: To find the derivative of the function f(x)=-5(5x^(2)+9x)^(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 Outer Function Derivative: First, let's identify the outer function and the inner function. The outer function is u^6 where u is the inner function, and the inner function is 5x^2 + 9x.Find Inner Function Derivative: Now, we will find the derivative of the outer function with respect to the inner function u. The derivative of u^6 with respect to u is 6u^5.Apply Chain Rule: Next, we will find the derivative of the inner function 5x^2 + 9x with respect to x. The derivative of 5x^2 is 10x, and the derivative of 9x is 9. So, the derivative of the inner function is 10x + 9.Simplify Expression: Now, we apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us: f'(x) = -5 * 6(5x^2 + 9x)^5 * (10x + 9).Simplify the expression by combining constants and applying the power rule. f'(x) = -30(5x^2 + 9x)^5 * (10x + 9).

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Given the function
f(x)=-5(5x^(2)+9x)^(6), find
f^(')(x) in any form.
Answer:
f^(')(x)=

Given the function $f(x)=-5\left(5 x^{2}+9 x\right)^{6}$ , 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)=-5\left(5 x^{2}+9 x\right)^{6}

, find

f^{\prime}(x)

in any form.

\newline

Answer:

f^{\prime}(x)=

Identify Functions: To find the derivative of the function $f(x)=-5(5x^{2}+9x)^{6}$ , 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 Outer Function Derivative: First, let's identify the outer function and the inner function. The outer function is $u^6$ where $u$ is the inner function, and the inner function is $5x^2 + 9x$ .
Find Inner Function Derivative: Now, we will find the derivative of the outer function with respect to the inner function $u$ . The derivative of $u^6$ with respect to $u$ is $6u^5$ .
Apply Chain Rule: Next, we will find the derivative of the inner function $5x^2 + 9x$ with respect to $x$ . The derivative of $5x^2$ is $10x$ , and the derivative of $9x$ is $9$ . So, the derivative of the inner function is $10x + 9$ .
Simplify Expression: Now, we apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us: $f'(x) = -5 \times 6(5x^2 + 9x)^5 \times (10x + 9)$ .
Simplify Expression: Now, we apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us: $f'(x) = -5 \times 6(5x^2 + 9x)^5 \times (10x + 9)$ . Simplify the expression by combining constants and applying the power rule. $f'(x) = -30(5x^2 + 9x)^5 \times (10x + 9)$ .