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provide solution for RD Sharma maths class 12 chapter Differentiation exercise  10.2 question 57

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Answer: \frac{1}{x^{2}-1}

Hint: you must know the rule of solving derivative of logarithm functions

Given: \log \sqrt{\frac{x-1}{x+1}}

Solution:

Let  y=\log \left(\frac{x-1}{x+1}\right)^{\frac{1}{2}}

\begin{aligned} &y=\frac{1}{2} \log \left(\frac{x-1}{x+1}\right) \\\\ &y=\frac{1}{2}\{\log (x-1)-\log (x+1)\} \end{aligned}

Differentiate with respect to x

\frac{d y}{d x}=\frac{1}{2}\left\{\frac{d}{d x}[\log (x-1)]-\frac{d}{d x}[\log (x+1)]\right\}

\begin{aligned} &\frac{d y}{d x}=\frac{1}{2}\left[\frac{1}{(x-1)}-\frac{1}{(x+1)}\right] \\\\ &\frac{d y}{d x}=\frac{1}{2}\left[\frac{x+1-(x-1)}{\left(x^{2}-1\right)}\right] \end{aligned}

\begin{aligned} &\frac{d y}{d x}=\frac{1}{2}\left[\frac{x+1-x+1}{\left(x^{2}-1\right)}\right] \\\\ &\frac{d y}{d x}=\frac{1}{2}\left[\frac{2}{\left(x^{2}-1\right)}\right] \\\\ &\frac{d y}{d x}=\frac{1}{\left(x^{2}-1\right)} \end{aligned}

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