# What is the derivative of sin (x/2)?

Apr 18, 2018

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

#### Explanation:

The Chain Rule, when applied to the sine, tells us that

$\frac{d}{\mathrm{dx}} \sin \left(u\right) = \cos u \cdot \frac{\mathrm{du}}{\mathrm{dx}}$, where $u$ is some function in terms of $x .$ Here, we see $u = \frac{x}{2} ,$ so

$\frac{d}{\mathrm{dx}} \sin \left(\frac{x}{2}\right) = \cos \left(\frac{x}{2}\right) \cdot \frac{d}{\mathrm{dx}} \left(\frac{x}{2}\right)$

$\frac{d}{\mathrm{dx}} \left(\frac{x}{2}\right) = \frac{1}{2} ,$ so we end up with

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

Apr 18, 2018

$\cos \frac{\frac{x}{2}}{2}$

#### Explanation:

use chain rule:
$\frac{d}{\mathrm{dx}} \left(\sin \left(\frac{x}{2}\right)\right) = \cos \left(\frac{x}{2}\right) \cdot \frac{d}{\mathrm{dx}} \left(\frac{x}{2}\right)$

(note that derivative of $\sin x$ is $\cos x$)

$\cos \left(\frac{x}{2}\right) \cdot \left(\frac{1}{2}\right)$
$= \cos \frac{\frac{x}{2}}{2}$