Example 5. Find the derivative of the function $y=\ln \sqrt{\frac{1+2 x}{1-2 x}}$.
## Solution.

$$
\begin{gathered}
y^{\prime}=\left[\ln \left(\frac{1+2 x}{1-2 x}\right)^{\frac{1}{2}}\right]^{\prime}=\left[\frac{1}{2} \ln \left(\frac{1+2 x}{1-2 x}\right)\right]^{\prime}= \\
=\frac{1}{2}[\ln (1+2 x)-\ln (1-2 x)]^{\prime}=\frac{1}{2}\left(\frac{2}{1+2 x}+\frac{2}{1-2 x}\right)= \\
=\left(\frac{1}{1+2 x}+\frac{1}{1-2 x}\right)=\frac{2}{1-4 x^{2}}
\end{gathered}
$$

Remark. If the expression under the logarithmic function to be differentiated contains a term that can be logarithmized (product, quotient, power, root), it is useful to perform logarithmization first.