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

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

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Quotient property of logarithms

Full solution

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 $f'(x) = -12\sin(6x)$ .

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