Find the derivative of the function $y = \ln \left( \cos ^ { 2 } x \right)$ and simplify your answer by using the trigonometric identities.
A) $y = 2 \tan x$
B) $y = \frac { 2 } { \cos x \sin x }$
C) $y = - 2 \tan x$
D) $y = 2 \cot x$
E) $y = - 2 \cot x$

Question

Quizplus · Accepted Answer

To find the derivative of $y = \ln(\cos^2x)$, we use the chain rule and trigonometric identities. First, rewrite the function as $y = 2\ln(\cos x)$. The derivative of $\ln(\cos x)$ with respect to $x$ is $-\tan x$ (using the chain rule and the derivative of $\ln x$, which is $1/x$, and the derivative of $\cos x$, which is $-\sin x$). Therefore, the derivative of $y$ is $2 \cdot (-\tan x) = -2\tan x$.

Find the Derivative of the Function y=ln⁡(cos⁡2x)y = \ln \left( \cos ^ { 2 } x \right)y=ln(cos2x)

Find the Derivative of the Function $y = \ln \left( \cos ^ { 2 } x \right)$