Question

Find the derivative of \( \log(\cos x) \): \[ \frac{d}{dx} \left( \log(\cos x) \right) \]

• A. \( \tan x \)
• B. \( -\tan x \)
• C. \( \cot x \)
• D. \( -\cot x \)

Hint:

When differentiating a logarithmic function with a composite argument, use the chain rule. In this case, \( \frac{d}{dx} \left( \log(\cos x) \right) = -\frac{\sin x}{\cos x} = -\tan x \).

Answer 1

The Correct Option is D

To differentiate \( \log(\cos x) \), we use the chain rule. The derivative of \( \log(u) \) is \( \frac{1}{u} \), and then we differentiate \( \cos x \) to get \( -\sin x \). Thus: \[ \frac{d}{dx} \left( \log(\cos x) \right) = \frac{1}{\cos x} \cdot (-\sin x) = -\frac{\sin x}{\cos x} \] This simplifies to: \[ -\tan x \] Therefore, the correct option is: \[ \boxed{-\tan x} \]