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Find the derivative of f(x)=cos(2ln(-4x-3)).

Find the derivative of $f(x)=\cos (2 \ln (-4 x-3))$ .

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Find derivatives of logarithmic functions

Full solution

Q. Find the derivative of

f(x)=\cos (2 \ln (-4 x-3))

.

Identify Outer Function: Identify the outer function and its derivative. $\newline$ The outer function is the cosine function, so we need to find the derivative of $\cos(u)$ , where $u$ is a function of $x$ . The derivative of $\cos(u)$ with respect to $u$ is $-\sin(u)$ .
Identify Inner Function: Identify the inner function and its derivative. $\newline$ The inner function is $2\ln(-4x-3)$ . To find its derivative, we use the chain rule. The derivative of $\ln(v)$ with respect to $v$ is $1/v$ , so the derivative of $2\ln(v)$ is $2/v$ . Here, $v = -4x-3$ , so the derivative of the inner function is $2/(-4x-3)$ .
Apply Chain Rule: Apply the chain rule. $\newline$ 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. So, we have: $\newline$ $f'(x) = -\sin(2\ln(-4x-3)) \cdot \left(\frac{d}{dx}\right)(2\ln(-4x-3))$
Differentiate Inner Function: Differentiate the inner function. $\newline$ We already found that the derivative of $2\ln(-4x-3)$ is $\frac{2}{(-4x-3)}$ . Now we need to differentiate $-4x-3$ with respect to $x$ , which is simply $-4$ . So the derivative of the inner function is $\frac{2}{(-4x-3)} \times -4$ .
Simplify Inner Function Derivative: Simplify the derivative of the inner function. $\newline$ Multiplying $\frac{2}{(-4x-3)}$ by $-4$ gives us $\frac{-8}{(-4x-3)}$ , which simplifies to $\frac{8}{(-4x-3)}$ because the negatives cancel out.
Combine Derivatives: Combine the derivatives to find the final derivative. $\newline$ Now we multiply the derivative of the outer function by the simplified derivative of the inner function: $\newline$ $f'(x) = -\sin(2\ln(-4x-3)) \cdot \left(\frac{8}{-4x-3}\right)$
Simplify Final Expression: Simplify the final expression. $\newline$ We can leave the final derivative in its current form, as it is already simplified: $\newline$ $f'(x) = -8\sin(2\ln(-4x-3))/(-4x-3)$

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