# Derivative of Cosine Function/Proof 4

## Theorem

$\map {D_x} {\cos x} = -\sin x$

## Proof

 $\displaystyle D_x \left({\cos x}\right)$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {\cos \left({x + h}\right) - \cos \left({x}\right)} h$ Definition of Derivative of Real Function at Point $\displaystyle$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {\cos \left({\left({x + \frac h 2}\right) + \frac h 2}\right) - \cos \left({\left({x + \frac h 2}\right) - \frac h 2}\right)} h$ $\displaystyle$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {-2 \sin \left({x + \frac h 2}\right) \sin \left({\frac h 2}\right)} h$ Simpson's Formula for Sine by Sine $\displaystyle$ $=$ $\displaystyle -\lim_{h \mathop \to 0} \sin \left({x + \frac h 2}\right) \lim_{h \mathop \to 0} \frac{\sin \left({\frac h 2}\right)} {\frac h 2}$ Multiple Rule for Limits of Functions and Product Rule for Limits of Functions $\displaystyle$ $=$ $\displaystyle -\sin \left({x}\right) \times 1$ Continuity of Sine and Limit of Sine of X over X $\displaystyle$ $=$ $\displaystyle -\sin \left({x}\right)$

$\blacksquare$