# What is the derivative of this function sin x^(1/2)?

Sep 7, 2016

cos(x^(1/2))/(2x^(1/2)

#### Explanation:

differentiate using the $\textcolor{b l u e}{\text{chain rule}}$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder }} \textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \times \frac{\mathrm{du}}{\mathrm{dx}}} \textcolor{w h i t e}{\frac{a}{a}} |}}} \ldots . \left(A\right)$

let $u = {x}^{\frac{1}{2}} \Rightarrow \frac{\mathrm{du}}{\mathrm{dx}} = \frac{1}{2} {x}^{- \frac{1}{2}}$

and $y = \sin \left(u\right) \Rightarrow \frac{\mathrm{dy}}{\mathrm{du}} = \cos \left(u\right)$

substitute these values into (A) changing u into terms of x

rArrdy/dx=cos(u)xx1/2x^(-1/2)=cos(x^(1/2))/(2x^(1/2)