Find a formula for the derivative of the function
![\[ f(x) = \frac{x}{1+\sqrt{x}}, \qquad x > 0. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-21276bf438af9ec3640f0c38353b90c8_l3.png "Rendered by QuickLaTeX.com")
Using the formula for the derivative of a rational power of 
and the quotient rule, we have
![\begin{align*} f'(x) &= \frac{1+\sqrt{x} - x\left(\frac{1}{2}x^{-\frac{1}{2}}\right)}{(1+\sqrt{x})^2}\\[8pt] &= \frac{2+\sqrt{x}}{2(1+\sqrt{x})^2}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-aa5a038561fb56d6061fe699cc305380_l3.png "Rendered by QuickLaTeX.com")
for 
.
Find a formula for the derivative of the function
Using the formula for the derivative of a rational power of and the quotient rule, we have
for .