Cos(X) = \frac{ e^{ix} + e^{-ix}}{2} \frac{d}{dx} Cos(X) = \frac{d}{dx} [ \frac{ e^{ix} + e^{-ix}}{2} ] \frac{1}{2} is a constant. \frac{d}{dx} Cos(X) = \frac{1}{2} [ \frac{d}{dx} e^{ix} + \frac{d}{dx} e^{-ix} ] = \frac{1}{2} [ i e^{ix} + ( -i e^{-ix} ] = \frac{1}{2} [ i e^{ix} – i e^{-ix} ] = \frac{i}{2} [ e^{ix} – e^{-ix} ] = \frac{i}{i}*\frac{i}{2} [ e^{ix} – e^{-ix} ] = -\frac{1}{2 i} [ e^{ix} – e^{-ix} ] = -Sin(X)