Find a formula for the derivative of the function
![\[ f(x) = \frac{1}{1+\sqrt{x}}, \qquad x > 0. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-a1ba2df3034bb0cf79038377315f196a_l3.png "Rendered by QuickLaTeX.com")
Using the quotient rule for derivatives, and the formula for the derivative of a rational power of 
, we have
![\[ f'(x) = \frac{-\frac{1}{2} x^{-\frac{1}{2}}}{(1+\sqrt{x})^2} = - \frac{1}{2 \sqrt{x} (1+\sqrt{x})^2}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-3a2a0b874fb3931e3ac086a339644c71_l3.png "Rendered by QuickLaTeX.com")
Find a formula for the derivative of the function
Using the quotient rule for derivatives, and the formula for the derivative of a rational power of , we have