Question

The value of \(\frac{d(\log \cos x)}{dx}\) will be

• A. \(\text{Tan}(x)\)
• B. \(-\text{Tan}(x)\)
• C. \(1/\cos x\)
• D. \(-1/\cos x\)

Hint:

Logarithmic differentiation of trigonometric functions often yields simple tangent or cotangent results.
Keep basic derivatives in mind: \(\frac{d}{dx}(\log \sin x) = \cot x\) and \(\frac{d}{dx}(\log \cos x) = -\tan x\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To find the derivative of a composite function, we must apply the chain rule of differentiation.

Step 2: Key Formula or Approach:
The chain rule states:
\[ \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \] Specifically, for logarithms:
\[ \frac{d}{dx} [\log(u)] = \frac{1}{u} \frac{du}{dx} \]

Step 3: Detailed Explanation:
Let the function be \(y = \log(\cos x)\).
Applying the chain rule with \(u = \cos x\):
\[ \frac{dy}{dx} = \frac{1}{\cos x} \cdot \frac{d}{dx}(\cos x) \] We know that the derivative of \(\cos x\) is \(-\sin x\):
\[ \frac{dy}{dx} = \frac{1}{\cos x} \cdot (-\sin x) \] \[ \frac{dy}{dx} = -\frac{\sin x}{\cos x} = -\tan x \]

Step 4: Final Answer:
The correct option is 2, which corresponds to \(-\text{Tan}(x)\).