Question Detail
Question
Find $\frac{dy}{dx}$, if $y = \log_e \cos x$.
Correct Answer
Option C is the correct answer.
Detailed Explanation
The notation $\log_e$ refers to the natural logarithm, so $y = \ln(\cos x)$.To differentiate this, we use the chain rule: $\frac{d}{dx} \ln(f(x)) = \frac{f'(x)}{f(x)}$.Here, $f(x) = \cos x$, so $f'(x) = -\sin x$.Therefore, $\frac{dy}{dx} = \frac{-\sin x}{\cos x}$.Since $\frac{\sin x}{\cos x} = \tan x$, we have:$\frac{dy}{dx} = -\tan x$.
Hint
Recall that $\log_e x = \ln x$. Apply the chain rule for differentiating $\ln(f(x))$. Which trigonometric identity relates $\sin x$ and $\cos x$?