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: Split the log
$\log\frac{x+3}{x} = \log(x+3) - \log x$.

Step 2: Inner derivative
$\frac{1}{x+3} - \frac{1}{x} = -\frac{3}{x(x+3)}$.

Step 3: Chain rule
Multiply by $\cos$ of the same argument: option (C).

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