Derivative of Cosine Function/Proof 2

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Theorem

$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$


Proof

\(\displaystyle \map {\frac \d {\d x} } {\cos x}\) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {\map \cos {x + h} - \cos x} h\) Definition of Derivative of Real Function at Point
\(\displaystyle \) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {\cos x \cos h - \sin x \sin h - \cos x} h\) Cosine of Sum
\(\displaystyle \) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {\cos x \cos h - \cos x} h + \lim_{h \mathop \to 0} \frac {-\sin x \sin h} h\) Sum Rule for Limits of Real Functions
\(\displaystyle \) \(=\) \(\displaystyle \cos x \lim_{h \mathop \to 0} \frac {\cos h - 1} h - \sin x \lim_{h \mathop \to 0} \frac {\sin h} h\) Multiple Rule for Limits of Real Functions
\(\displaystyle \) \(=\) \(\displaystyle \cos x \times 0 - \sin x \times 1\) Limit of (Cosine (X) - 1) over X and Limit of Sine of X over X
\(\displaystyle \) \(=\) \(\displaystyle -\sin x\)

$\blacksquare$