Derivative of Cosine Function/Proof 2

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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
\(\displaystyle \) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {\cos \left({x}\right) \cos \left({h}\right) - \sin \left({x}\right) \sin \left({h}\right) - \cos \left({x}\right)} h\) Cosine of Sum
\(\displaystyle \) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {\cos \left({x}\right) \cos \left({h}\right) - \cos \left({x}\right)} h + \lim_{h \mathop \to 0} \frac{- \sin \left({x}\right) \sin \left({h}\right)} h\) Sum Rule for Limits of Functions
\(\displaystyle \) \(=\) \(\displaystyle \cos \left({x}\right) \ \lim_{h \mathop \to 0} \frac {\cos \left({h}\right) - 1} h - \sin \left({x}\right) \ \lim_{h \mathop \to 0} \frac {\sin \left({h}\right)} h\) Multiple Rule for Limits of Functions
\(\displaystyle \) \(=\) \(\displaystyle \cos \left({x}\right) \times 0 - \sin \left({x}\right) \times 1\) Limit of (Cosine (X) - 1) over X and Limit of Sine of X over X
\(\displaystyle \) \(=\) \(\displaystyle - \sin \left({x}\right)\)

$\blacksquare$