Derivative of Sine Function/Proof 4

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Theorem

$D_x \left({\sin x}\right) = \cos x$


Proof


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

$\blacksquare$