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)

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