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Compute the derivative of x log(x+ (1+x2)1/2) – (1+x2)1/2

by RoRi

Compute the derivative of

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

Using the chain rule and the formula for the derivative of the logarithm we can compute directly,

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

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