Given the function f(x)=-3(4x^(2)+6x)^(6), find f^(')(x) in any form. Answer: f^(')(x)=

Identify Functions: To find the derivative of the function f(x) = -3(4x^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.Derivative of Outer Function: First, let's identify the outer function and the inner function. The outer function is u^6 where u = 4x^2 + 6x, and the inner function is 4x^2 + 6x. We will need to take the derivative of both.Derivative of Inner Function: The derivative of the outer function u^6 with respect to u is 6u^5. We will later substitute u with 4x^2 + 6x.Apply Chain Rule: The derivative of the inner function 4x^2 + 6x with respect to x is 8x + 6. This is obtained by using the power rule for each term separately.Simplify Expression: 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) = 6 * (-3) * (4x^2 + 6x)^5 * (8x + 6).Final Derivative: Simplify the expression by multiplying the constants and combining like terms: f'(x) = -18 * (4x^2 + 6x)^5 * (8x + 6).The final form of the derivative is: f'(x) = -18(4x^2 + 6x)^5(8x + 6).

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Given the function
f(x)=-3(4x^(2)+6x)^(6), find
f^(')(x) in any form.
Answer:
f^(')(x)=

Given the function $f(x)=-3\left(4 x^{2}+6 x\right)^{6}$ , find $f^{\prime}(x)$ in any form. $\newline$ Answer: $f^{\prime}(x)=$

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Q. Given the function

f(x)=-3\left(4 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 $f(x) = -3(4x^2 + 6x)^6$ , 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.
Derivative of Outer Function: First, let's identify the outer function and the inner function. The outer function is $u^6$ where $u = 4x^2 + 6x$ , and the inner function is $4x^2 + 6x$ . We will need to take the derivative of both.
Derivative of Inner Function: The derivative of the outer function $u^6$ with respect to $u$ is $6u^5$ . We will later substitute $u$ with $4x^2 + 6x$ .
Apply Chain Rule: The derivative of the inner function $4x^2 + 6x$ with respect to $x$ is $8x + 6$ . This is obtained by using the power rule for each term separately.
Simplify Expression: 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: $\newline$ $f'(x) = 6 \times (-3) \times (4x^2 + 6x)^5 \times (8x + 6)$ .
Final Derivative: Simplify the expression by multiplying the constants and combining like terms: $f'(x) = -18 \times (4x^2 + 6x)^5 \times (8x + 6)$ .
Final Derivative: Simplify the expression by multiplying the constants and combining like terms: $f'(x) = -18 \times (4x^2 + 6x)^5 \times (8x + 6)$ . The final form of the derivative is: $f'(x) = -18(4x^2 + 6x)^5(8x + 6)$ .