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Derivative of f(x)=sin x

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The derivative of $ f(x)=\sin x $ can be determined with the aid of the addition theorem. From

$\displaystyle \sin(t\pm h/2)=\sin t\cos(h/2) \pm \cos t \sin(h/2) $

with $ t=x+h/2$, it follows that the difference quotient equals

$\displaystyle \frac{\sin(x+h)-\sin x }{h} = \frac{\sin\big((x+h/2)+h/2\big)-\sin\big((x+h/2)-h/2\big)}{h} = \frac{2\cos(x+h/2)\sin(h/2)}{h}\,. $

Since

$\displaystyle \lim_{h \to0} \frac{2\sin(h/2)}{h}= \lim_{h \to0} \frac{\sin(h/2)}{h/2}=1\,, $

the right-hand side tends to $ \cos x $ for $ h\to0$ .

see also:

  automatisch erstellt am 14.  6. 2016