# Derivative of Sine Function/Proof 4

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