Question Detail
Question
Find $\frac{dy}{dx}$, if $y = \log_e (\cos x)$.
Correct Answer
Option C is the correct answer.
Detailed Explanation
To find the derivative of $y = \log_e (\cos x)$, also written as $y = \ln(\cos x)$, we use the chain rule.The derivative of $\ln(u)$ is $\frac{1}{u} \frac{du}{dx}$.Here, $u = \cos x$.So, $\frac{du}{dx} = \frac{d}{dx}(\cos x) = -\sin x$.Therefore,$\frac{dy}{dx} = \frac{1}{\cos x} \cdot (-\sin x) = -\frac{\sin x}{\cos x} = -\tan x$.
Hint
Remember the chain rule for derivatives involving composite functions, and the derivatives of $\ln x$ and $\cos x$.