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

$\blacksquare$