Blog: `y = ln sqrt((x + 1)/(x - 1)))` Find the derivative of the function.

$\displaystyle{y}={\ln{\sqrt{{\frac{{{x}+{1}}}{{{x}-{1}}}}}}}$

$\displaystyle{y}={{\ln{{\left({x}+{1}\right)}}}}^{{\frac{{1}}{{2}}}}-{{\ln{{\left({x}-{1}\right)}}}}^{{\frac{{1}}{{2}}}}$

$\displaystyle{y}=\frac{{1}}{{2}}{\ln{{\left({x}+{1}\right)}}}-\frac{{1}}{{2}}{\ln{{\left({x}-{1}\right)}}}$

$\displaystyle{y}'=\frac{{1}}{{{2}{\left({x}+{1}\right)}}}-\frac{{1}}{{{2}{\left({x}-{1}\right)}}}$

$\displaystyle{y}'=\frac{{{\left({x}-{1}\right)}-{\left({x}+{1}\right)}}}{{{2}{\left({x}+{1}\right)}{\left({x}-{1}\right)}}}$

$\displaystyle{y}'=\frac{{{x}-{1}-{x}-{1}}}{{{2}{\left({x}+{1}\right)}{\left({x}-{1}\right)}}}$

$\displaystyle{y}'=-\frac{{1}}{{{\left({x}+{1}\right)}{\left({x}-{1}\right)}}}$

The derivative of the function y is $\displaystyle-\frac{{1}}{{{\left({x}+{1}\right)}{\left({x}-{1}\right)}}}.$