Discussion
If \(y = \log\left(\sqrt{\frac{x - 1}{x + 2}}\right)\) then \(\frac{dy}{dx} =\)
Answer: B
\(\frac{dy}{dx} = \frac{3}{2(x - 1)(x + 2)}\).
Simplify with log laws before differentiating:
\[ \begin{aligned} y &= \tfrac{1}{2}[\log(x - 1) - \log(x + 2)] \\ \frac{dy}{dx} &= \frac{1}{2}\left[\frac{1}{x - 1} - \frac{1}{x + 2}\right] \\ &= \frac{1}{2} \cdot \frac{3}{(x - 1)(x + 2)} \end{aligned} \]
Forgetting the \(\frac{1}{2}\) from the square root gives \(\frac{3}{(x - 1)(x + 2)}\), and forgetting the 3 from combining the fractions gives \(\frac{1}{2(x - 1)(x + 2)}\).
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