Question

$\frac{d}{dx}\left(\log\left(\sqrt{x+\sqrt{x^{2}+a^{2}}}\right)\right) = $

• A. $\sqrt{x^{2} +a^{2}}$
• B. $\frac{1}{\sqrt{x^{2} +a^{2}}}$
• C. $\frac{1}{ 2 \sqrt{x^{2} +a^{2}}}$
• D. $\frac{1}{2(x + \sqrt{x^{2} +a^{2}} )}$

Answer 1

Answer: (C)

$ \frac{d}{d x}\left(\log \left(\sqrt{x+\sqrt{x^{2}+a^{2}}}\right)\right)$
$= \frac{1}{\sqrt{x+\sqrt{x^{2}+a^{2}}}} \times\left(\frac{1}{2 \sqrt{x+\sqrt{x^{2}+a^{2}}}}\right) $
$ \times \left( 1 + \frac{1}{2} \cdot \frac{2x}{\sqrt{x^2 + a^2}} \right)$
$= \frac{\left(\sqrt{x^{2}+a^{2}}+x\right)}{2 \sqrt{x^{2}+a^{2}}\left(x+\sqrt{x^{2}+a^{2}}\right)}$
$=\frac{1}{2 \sqrt{x^{2}+a^{2}}}$