Question

The derivative of \(sin(log(\frac{x+3}{x}))\) is

• A. \(\frac{3}{x+3}cos(log(\frac{x+3}{x}))\)
• B. \(\frac{3}{x(x+3)}cos(log(\frac{x+3}{x}))\)
• C. \(\frac{-3}{x(x+3)}cos(log(\frac{x+3}{x}))\)
• D. \(\frac{-1}{x(x+3)}cos(log(\frac{x+3}{x}))\)

Hint:

Apply the chain rule three times and simplify the derivative of the inner fraction.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Let \(u = \log\frac{x + 3}{x}\). Then \(\frac{d}{dx}\sin u = \cos u \cdot \frac{du}{dx}\).

Step 2: Differentiate u
\(u = \log(x + 3) - \log x\), so
\[ \frac{du}{dx} = \frac{1}{x + 3} - \frac1x = \frac{x - (x + 3)}{x(x + 3)} = \frac{-3}{x(x + 3)} \]

Step 3: Combine
\[ \frac{d}{dx}\sin\left(\log\frac{x + 3}{x}\right) = \frac{-3}{x(x + 3)}\cos\left(\log\frac{x + 3}{x}\right) \]
Option (B) has the sign reversed, (A) uses only the first term, and (D) has the wrong constant.

Final Answer:
The derivative is option (C). \[ \boxed{\frac{-3}{x(x+3)}\cos\left(\log\frac{x+3}{x}\right)} \]