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Compute the derivative of the given function

by RoRi

Compute the derivative of the function

    \[ f(x) = \frac{1}{\sqrt{1+x^2} \left( x + \sqrt{1+x^2} \right)}. \]

First, we simplify the expression for f,

    \begin{align*} f(x) &= \frac{1}{\sqrt{1+x^2} \left( \sqrt{1+x^2} + x \right)} \\[8pt] &= \frac{\sqrt{1+x^2} - x}{\sqrt{1+x^2} \left( \sqrt{1+x^2} + x \right) \left( \sqrt{1+x^2} - x \right)} \\[8pt] &= \frac{\sqrt{1+x^2} - x}{\sqrt{1+x^2} (1 + x^2 - x^2)} \\ &= 1 - \frac{x}{\sqrt{1+x^2}}. \end{align*}

Now, we take the derivative. (Of course, you could take the derivative directly from the given form of f, I just find this form somewhat easier, but it was not so obvious we could simplify f this way.)

    \begin{align*} f'(x) &= - \left( \frac{ \sqrt{1+x^2} - \frac{1}{2} \frac{x}{\sqrt{1+x^2}} (2x) }{1+x^2}\right) \\[8pt] &= - \left( \frac{\sqrt{1+x^2} - \frac{x^2}{\sqrt{1+x^2}}}{1+x^2}\right) \\[8pt] &= - \left( \frac{1+x^2 - x^2}{(1+x^2)^{\frac{3}{2}}}\right) \\[8pt] &= \frac{-1}{(1+x^2)^{\frac{3}{2}}}. \end{align*}

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