Sine

\sin(x)

Derivative

\begin{aligned} \frac{d (\sin x)}{d x} & = lim_{h \to 0} (\frac{\sin (x + h) - \sin x}{h}) \\ = lim_{h \to 0} (\frac{\sin x \cos h + \sin h \cos x - \sin x}{h}) \\ = lim_{h \to 0} (\frac{(\cos h - 1) \sin x}{h} + \frac{\sin h}{h} \cos x) \\ = lim_{h \to 0} (\frac{\cos h - 1}{h}) \sin x + lim_{h \to 0} (\frac{\sin h}{h}) \cos x \\ = lim_{h \to 0} (\frac{\cos^{2} h - 1}{h (\cos h + 1)}) + lim_{h \to 0} (\frac{\sin h}{h}) \cos x \\ = lim_{h \to 0} (\frac{- \sin^{2} h}{h (\cos h + 1)}) + lim_{h \to 0} (\frac{\sin h}{h}) \cos x \\ = 1 (0) + 1 (\cos x) \\ = \cos x \end{aligned}