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Compute the derivative of the given function

Compute the derivative of the function

    \[ f(x) = x \sqrt{1+x^2}. \]


Using the chain rule,

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

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