Question

Differentiate

$\log \sqrt{\frac{x-1}{x+1}}$

Answer 1

$$\begin{gathered} \mathrm{Let}y=\log \sqrt{\frac{x-1}{x+1}} \\ \Rightarrow y=\log {\left(\frac{x-1}{x+1}\right)}^{\frac{1}{2}} \\ \Rightarrow y=\frac{1}{2}\log \left(\frac{x-1}{x+1}\right) \\ \Rightarrow y=\frac{1}{2}\left[\log \left(x-1\right)-\log \left(x+1\right)\right] \end{gathered}$$
Differentiate it with respect to x
$$\begin{gathered} \frac{\mathit{d}y}{\mathit{d}x}=\frac{1}{2}\left[\frac{d}{dx}\left\{\log \left(x-1\right)\right\}-\frac{d}{dx}\left\{\log \left(x+1\right)\right\}\right] \\ =\frac{1}{2}\left(\frac{1}{x-1}-\frac{1}{x+1}\right) \\ =\frac{1}{2}\left(\frac{2}{x^2-1}\right) \\ =\frac{1}{x^2-1} \\ \mathrm{So},\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=\frac{1}{x^2-1} \end{gathered}$$