Question

Find \[ \frac{dy}{dx} \] if \[ y=\log(\sin x). \]

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

Hint:

Whenever a logarithm contains another function inside it, use the formula \(\frac{d}{dx}(\log u)=\frac{u'}{u}\).

Answer 1

The Correct Option is B

Concept: The function is a composite function because \(\log\) contains another function \(\sin x\) inside it. Therefore, we use the Chain Rule: \[ \frac{d}{dx}(\log u) = \frac{1}{u}\cdot\frac{du}{dx} \]

Step 1: Identify the inner function Given \[ y=\log(\sin x) \] Let \[ u=\sin x \] Then \[ y=\log u \]

Step 2: Differentiate using the chain rule Applying \[ \frac{d}{dx}(\log u) = \frac1u\frac{du}{dx} \] we get \[ \frac{dy}{dx} = \frac1{\sin x}\cdot\frac{d}{dx}(\sin x) \] \[ = \frac1{\sin x}\cdot\cos x \]

Step 3: Simplify the expression \[ \frac{\cos x}{\sin x} = \cot x \] Therefore, \[ \frac{dy}{dx} = \cot x \] Final Answer: \[ \boxed{\cot x} \]