### Theorem #### Proof [[Angle Sum-or-Difference Identities]] [[Table of Common limits]] $\begin{aligned} \frac{d}{d x}(\cos x) &=\lim _{h \rightarrow 0} \frac{\cos (x+h)-\cos x}{h} \\ &=\lim _{h \rightarrow 0} \frac{(\cos x \cos h-\sin x \sin h)-\cos x}{h} \\ &=\lim _{h \rightarrow 0} \frac{\cos x(\cos h-1)-\sin x \sin h}{h} \\ &=\lim _{h \rightarrow 0} \cos x \cdot \frac{\cos h-1}{h}-\lim _{h \rightarrow 0} \sin x \cdot \frac{\sin h}{h} \\ &=\cos x \cdot \lim _{h \rightarrow 0} \frac{\cos h-1}{h}-\sin x \cdot \lim _{h \rightarrow 0} \frac{\sin h}{h} \\ &=\cos x \cdot 0-\sin x \cdot 1 \\ &=-\sin x . \end{aligned}$