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Find a formula for the derivative of a given function

Find a formula for the derivative of the function

    \[ f(x) = \frac{\sqrt{x}}{1+x} \qquad x > 0. \]


Using the formula for the derivative of a rational power of x and the quotient rule for derivatives, we have

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

for x > 0.

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