d/dx Sin(x)

Sin(x) = \frac{e^{ix} – e^{-ix}}{2i}

\frac{d}{dx} Sin(x) = \frac{d}{dx} ( \frac{e^{ix} – e^-{ix}}{2i} )

\frac{1}{2i} is  a  constant

\frac{1}{2i} \frac{d}{dx} ( e^{ix} – e^{-ix} )

\frac{1}{2i} [\frac{d}{dx} ( e^{ix} ) – \frac{d}{dx} (e^{-ix} ) ]

\frac{1}{2i} [ i e^{ix} – ( -i e^{-ix} ) ]

\frac{1}{2i} [ ie^{ix} + ie^{-ix}]

\frac{i}{2i} [ e^{ix} + e^{-ix}]

\frac{1}{2} [ e^{ix} + e^{-ix}] = Cos(x)