What is the derivative of $sin (\frac{x}{2})$ $sin (x/2)$ ?

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What is the derivative of $sin (\frac{x}{2})$ $sin (x/2)$ ?

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Answer 1 · 2018-04-18T03:36:03

$\frac{d}{d x} sin (\frac{x}{2}) = \frac{1}{2} cos (\frac{x}{2})$ $d/dxsin(x/2)=1/2cos(x/2)$

Explanation:

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

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

$\frac{d}{d x} sin (\frac{x}{2}) = cos (\frac{x}{2}) \cdot \frac{d}{d x} (\frac{x}{2})$ $d/dxsin(x/2)=cos(x/2)*d/dx(x/2)$

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

$\frac{d}{d x} sin (\frac{x}{2}) = \frac{1}{2} cos (\frac{x}{2})$ $d/dxsin(x/2)=1/2cos(x/2)$

Answer 2 · 2018-04-18T03:36:39

$\frac{cos (\frac{x}{2})}{2}$ $cos(x/2)/2$

Explanation:

use chain rule:
$\frac{d}{d x} (sin (\frac{x}{2})) = cos (\frac{x}{2}) \cdot \frac{d}{d x} (\frac{x}{2})$ $d/dx(sin(x/2))=cos(x/2)*d/dx(x/2)$

(note that derivative of $sin x$ $sinx$ is $cos x$ $cosx$ )

$cos (\frac{x}{2}) \cdot (\frac{1}{2})$ $cos(x/2)*(1/2)$
$= \frac{cos (\frac{x}{2})}{2}$ $=cos(x/2)/2$

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