ddxx+a2+x2=1x+a2+x21+1 2x2a2+x2=1x+a2+x2a2+x2+xa2+x2=1a2+x2

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

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$\frac{1}{(x + \sqrt{a^{2} + x^{2}})}$

$\frac{x}{\sqrt{a^{2} + x^{2}}}$

$\frac{1}{x (x + \sqrt{a^{2} + x^{2}})}$

$\frac{1}{\sqrt{a^{2} + x^{2}}}$

answer is D.

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Detailed Solution

$\frac{d}{d x} (x + \sqrt{a^{2} + x^{2}}) = \frac{1}{x + \sqrt{a^{2} + x^{2}}} (1 + \frac{12 x}{2 \sqrt{a^{2} + x^{2}}})$

$\begin{array}{l} = \frac{1}{x + \sqrt{a^{2} + x^{2}}} \frac{(\sqrt{a^{2} + x^{2}} + x)}{\sqrt{a^{2} + x^{2}}} \\ = \frac{1}{\sqrt{a^{2} + x^{2}}} \end{array}$

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