Derivative of Cosine Function/Proof 2

Theorem

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

Proof

\(\ds \map {\frac \d {\d x} } {\cos x}\)

\(=\)

\(\ds \lim_{h \mathop \to 0} \frac {\map \cos {x + h} - \cos x} h\)

Definition of Derivative of Real Function at Point

\(\ds \)

\(=\)

\(\ds \lim_{h \mathop \to 0} \frac {\cos x \cos h - \sin x \sin h - \cos x} h\)

Cosine of Sum

\(\ds \)

\(=\)

\(\ds \lim_{h \mathop \to 0} \frac {\cos x \cos h - \cos x} h + \lim_{h \mathop \to 0} \frac {-\sin x \sin h} h\)

Sum Rule for Limits of Real Functions

\(\ds \)

\(=\)

\(\ds \cos x \lim_{h \mathop \to 0} \frac {\cos h - 1} h - \sin x \lim_{h \mathop \to 0} \frac {\sin h} h\)

Multiple Rule for Limits of Real Functions

\(\ds \)

\(=\)

\(\ds \cos x \times 0 - \sin x \times 1\)

Limit of $\dfrac {\cos x - 1} x$ at Zero and Limit of $\dfrac {\sin x} x$ at Zero

\(\ds \)

\(=\)

\(\ds -\sin x\)

$\blacksquare$