Derivative of Cosine Function/Proof 4

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Theorem

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


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
\(\displaystyle \) \(=\) \(\displaystyle -\sin \left({x}\right)\)

$\blacksquare$