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

Compute the derivative of

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


Using the chain rule we can directly compute,

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

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