Question
Differentiate the following w. r. t. x.$y=\log \left[\frac{x+\sqrt{x^2+a^2}}{\sqrt{x^2+a^2}-x}\right]$
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Answer
$
\begin{aligned}
y=\log \left[\frac{x+\sqrt{x^2+a^2}}{\sqrt{x^2+a^2}-x}\right] & =\log \left[\frac{\sqrt{x^2+a^2}+x}{\sqrt{x^2+a^2}-x} \times \frac{\sqrt{x^2+a^2}+x}{\sqrt{x^2+a^2}+x}\right] \\
& =\log \left[\frac{\left(\sqrt{x^2+a^2}+x\right)^2}{x^2+a^2-x^2}\right] \\
& =\log \left[\frac{\left(\sqrt{x^2+a^2}+x\right)^2}{a^2}\right] \\
& =\log \left(\sqrt{x^2+a^2}+x\right)^2-\log \left(a^2\right) \\
\therefore \quad & =2 \log \left(\sqrt{x^2+a^2}+x\right)-\log \left(a^2\right)
\end{aligned}
$
Differentiate $w, r, t . x$$\begin{aligned} \frac{d y}{d x} & =\frac{d}{d x}\left[2 \log \left(\sqrt{x^2+a^2}+x\right)-\log \left(a^2\right)\right] \\ & =2 \frac{d}{d x}\left[\log \left(\sqrt{x^2+a^2}+x\right)\right]-\frac{d}{d x}\left[\log \left(a^2\right)\right] \\ & =2 \times \frac{1}{\sqrt{x^2+a^2}+x} \cdot \frac{d}{d x}\left[\sqrt{x^2+a^2}+x\right]-0 \\ & =\frac{2}{\sqrt{x^2+a^2}+x} \cdot\left[\frac{1}{2 \sqrt{x^2+a^2}} \cdot \frac{d}{d x}\left(x^2+a^2\right)+1\right] \\ & =\frac{2}{\sqrt{x^2+a^2}+x} \cdot\left[\frac{1}{2 \sqrt{x^2+a^2}}(2 x)+1\right] \\ & =\frac{2}{\sqrt{x^2+a^2}+x} \cdot\left[\frac{x+\sqrt{x^2+a^2}}{\sqrt{x^2+a^2}}\right] \\ \frac{d y}{d x} & =\frac{2}{\sqrt{x^2+a^2}}\end{aligned}$
\begin{aligned}
y=\log \left[\frac{x+\sqrt{x^2+a^2}}{\sqrt{x^2+a^2}-x}\right] & =\log \left[\frac{\sqrt{x^2+a^2}+x}{\sqrt{x^2+a^2}-x} \times \frac{\sqrt{x^2+a^2}+x}{\sqrt{x^2+a^2}+x}\right] \\
& =\log \left[\frac{\left(\sqrt{x^2+a^2}+x\right)^2}{x^2+a^2-x^2}\right] \\
& =\log \left[\frac{\left(\sqrt{x^2+a^2}+x\right)^2}{a^2}\right] \\
& =\log \left(\sqrt{x^2+a^2}+x\right)^2-\log \left(a^2\right) \\
\therefore \quad & =2 \log \left(\sqrt{x^2+a^2}+x\right)-\log \left(a^2\right)
\end{aligned}
$
Differentiate $w, r, t . x$$\begin{aligned} \frac{d y}{d x} & =\frac{d}{d x}\left[2 \log \left(\sqrt{x^2+a^2}+x\right)-\log \left(a^2\right)\right] \\ & =2 \frac{d}{d x}\left[\log \left(\sqrt{x^2+a^2}+x\right)\right]-\frac{d}{d x}\left[\log \left(a^2\right)\right] \\ & =2 \times \frac{1}{\sqrt{x^2+a^2}+x} \cdot \frac{d}{d x}\left[\sqrt{x^2+a^2}+x\right]-0 \\ & =\frac{2}{\sqrt{x^2+a^2}+x} \cdot\left[\frac{1}{2 \sqrt{x^2+a^2}} \cdot \frac{d}{d x}\left(x^2+a^2\right)+1\right] \\ & =\frac{2}{\sqrt{x^2+a^2}+x} \cdot\left[\frac{1}{2 \sqrt{x^2+a^2}}(2 x)+1\right] \\ & =\frac{2}{\sqrt{x^2+a^2}+x} \cdot\left[\frac{x+\sqrt{x^2+a^2}}{\sqrt{x^2+a^2}}\right] \\ \frac{d y}{d x} & =\frac{2}{\sqrt{x^2+a^2}}\end{aligned}$
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