# How do you find the derivative of cos[sqrt (x)]?

$\frac{d}{\mathrm{dx}} \left(\sqrt{x}\right) = \frac{d}{\mathrm{dx}} \left({x}^{\frac{1}{2}}\right) = \frac{1}{2} {x}^{\frac{1}{2} - 1} = \frac{1}{2} {x}^{\frac{1}{2} - \frac{2}{2}} = \frac{1}{2} {x}^{- \frac{1}{2}} = \frac{1}{2 {x}^{\frac{1}{2}}} = \frac{1}{2 \sqrt{x}}$
$\frac{d}{\mathrm{dx}} \left(\cos \left(\sqrt{x}\right)\right) = - \sin \left(\sqrt{x}\right) \frac{d}{\mathrm{dx}} \left(\sqrt{x}\right)$
$= - \sin \left(\sqrt{x}\right) \left(\frac{1}{2 \sqrt{x}}\right)$