Question

If \(y = \log \sqrt{\frac{x-1}{x+2}}\), then \(\frac{dy}{dx}\) is

• A. $\frac{1}{2(x-1)(x+2)}$
• B. $\frac{3}{2(x-1)(x+2)}$
• C. $\frac{3}{(x-1)(x+2)}$
• D. $\frac{1}{(x-1)(x+2)}$
• E. $\frac{1}{3(x-1)(x+2)}$

Hint:

Convert roots into logarithmic powers to simplify differentiation.

Answer 1

The Correct Option is B

Concept: Use logarithmic differentiation: \[ \log\sqrt{A} = \frac{1}{2}\log A \]

Step 1: Simplify expression
\[ y = \frac{1}{2} \log\left(\frac{x-1}{x+2}\right) \]

Step 2: Differentiate
\[ \frac{dy}{dx} = \frac{1}{2}\left(\frac{1}{x-1} - \frac{1}{x+2}\right) \]

Step 3: Simplify
\[ = \frac{1}{2} \cdot \frac{(x+2)-(x-1)}{(x-1)(x+2)} \] \[ = \frac{1}{2} \cdot \frac{3}{(x-1)(x+2)} = \frac{3}{2(x-1)(x+2)} \] Final Conclusion:
Option (B)