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 · 2016-06-21T11:31:38

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

Explanation:

if you know that $\frac{d}{d x} sin x = cos x$ $d/dx sin x = cos x$ then you can use the chain rule so by letting $u = \frac{x}{2}$ $u = x / 2$ you have $\frac{d}{d u} sin u = cos u$ $d/(du) sin u = cos u$

and $\frac{d u}{d x} = \frac{d}{d x} (\frac{x}{2}) = \frac{1}{2}$ $(du)/dx = (d)/dx (x/2)= 1/2$

$s o \frac{d}{d u} (sin u) \times \frac{d u}{d x} = cos u \times \frac{1}{2} = \frac{1}{2} cos (\frac{x}{2})$ $so d/{du} (sin u) \times (du)/(dx) = cos u \times 1/2 = 1/2 cos (x/2)$

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

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