Given the function f(x)=2cos(4-4x), find f^(')(x).

Identify Function: Identify the function to differentiate. We are given the function f(x) = 2cos(4-4x) and we need to find its derivative, which is denoted by 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 2cos(u) and the inner function is u = 4-4x.Differentiate Outer Function: Differentiate the outer function with respect to the inner function. The derivative of cos(u) with respect to u is -sin(u). Therefore, the derivative of 2cos(u) with respect to u is -2sin(u).Differentiate Inner Function: Differentiate the inner function with respect to x. The derivative of u = 4-4x with respect to x is -4, since the derivative of a constant is 0 and the derivative of -4x is -4.Apply Chain Rule: Apply the chain rule by multiplying the derivatives from steps 3 and 4. We multiply the derivative of the outer function, -2sin(u), by the derivative of the inner function, -4, to get the derivative of the composite function. This gives us f'(x) = -2sin(4-4x) * (-4).Simplify Derivative: Simplify the expression for the derivative. Multiplying -2 by -4 gives us 8, so f'(x) = 8sin(4-4x).

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Given the function $\newline$ $f(x)=2\cos(4-4x)$ , find $\newline$ $f^{\prime}(x)$ .

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Q. Given the function

\newline

f(x)=2\cos(4-4x)

, find

\newline

f^{\prime}(x)

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $f(x) = 2\cos(4-4x)$ and we need to find its derivative, which is denoted by $f'(x)$ .
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 $2\cos(u)$ and the inner function is $u = 4-4x$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function. $\newline$ The derivative of $\cos(u)$ with respect to $u$ is $-\sin(u)$ . Therefore, the derivative of $2\cos(u)$ with respect to $u$ is $-2\sin(u)$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The derivative of $u = 4-4x$ with respect to $x$ is $-4$ , since the derivative of a constant is $0$ and the derivative of $-4x$ is $-4$ .
Apply Chain Rule: Apply the chain rule by multiplying the derivatives from steps $3$ and $4$ . We multiply the derivative of the outer function, $-2\sin(u)$ , by the derivative of the inner function, $-4$ , to get the derivative of the composite function. This gives us $f'(x) = -2\sin(4-4x) \cdot (-4)$ .
Simplify Derivative: Simplify the expression for the derivative. Multiplying $-2$ by $-4$ gives us $8$ , so $f'(x) = 8\sin(4-4x)$ .