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provide solution for RD Sharma maths class 12 chapter Differentiation exercise  10.2 question 61

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Answer:   Proved

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

Given:   \mathrm{y}=\log \left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)

Prove   \frac{d y}{d x}=\frac{x-1}{2 x(x+1)}


Let  \mathrm{y}=\log \left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)

Differentiate with respect to x

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

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

∴ Proved


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