Find (d)/(dx)(2cos 6x) Answer:

Identify Function: Identify the function to differentiate. We need to find the derivative of the function f(x) = 2cos(6x) with respect to 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. Let's denote the outer function as g(u) = 2cos(u) and the inner function as u(x) = 6x.Differentiate Outer Function: Differentiate the outer function g(u) with respect to u. The derivative of g(u) = 2cos(u) with respect to u is g'(u) = -2sin(u), since the derivative of cos(u) is -sin(u).Differentiate Inner Function: Differentiate the inner function u(x) with respect to x. The derivative of u(x) = 6x with respect to x is u'(x) = 6, since the derivative of x with respect to x is 1 and we have a constant multiple of 6.Apply Chain Rule Derivative: Apply the chain rule using the derivatives from steps 3 and 4. The derivative of f(x) with respect to x is f'(x) = g'(u) * u'(x) = -2sin(u) * 6, where u = 6x.Substitute u: Substitute u = 6x back into the derivative. Substituting u = 6x into the derivative, we get f'(x) = -2sin(6x) * 6.Simplify Derivative: Simplify the derivative. Simplifying the expression, we get f'(x) = -12sin(6x).

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Find $\frac{d}{d x}(2 \cos 6 x)$ $\newline$ Answer:

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Q. Find

\frac{d}{d x}(2 \cos 6 x)

\newline

Answer:

Identify Function: Identify the function to differentiate. $\newline$ We need to find the derivative of the function $f(x) = 2\cos(6x)$ with respect to $x$ .
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. $\newline$ Let's denote the outer function as $g(u) = 2\cos(u)$ and the inner function as $u(x) = 6x$ .
Differentiate Outer Function: Differentiate the outer function $g(u)$ with respect to $u$ . The derivative of $g(u) = 2\cos(u)$ with respect to $u$ is $g'(u) = -2\sin(u)$ , since the derivative of $\cos(u)$ is $-\sin(u)$ .
Differentiate Inner Function: Differentiate the inner function $u(x)$ with respect to $x$ . The derivative of $u(x) = 6x$ with respect to $x$ is $u'(x) = 6$ , since the derivative of $x$ with respect to $x$ is $1$ and we have a constant multiple of $6$ .
Apply Chain Rule Derivative: Apply the chain rule using the derivatives from steps $3$ and $4$ . $\newline$ The derivative of $f(x)$ with respect to $x$ is $f'(x) = g'(u) \cdot u'(x) = -2\sin(u) \cdot 6$ , where $u = 6x$ .
Substitute $u$ : Substitute $u = 6x$ back into the derivative. $\newline$ Substituting $u = 6x$ into the derivative, we get $f'(x) = -2\sin(6x) \times 6$ .
Simplify Derivative: Simplify the derivative. $\newline$ Simplifying the expression, we get $f'(x) = -12\sin(6x)$ .