Why is the derivative of the complex definition of sin(x) equal to icos(x)?

Question

Using $\sin x = \frac{e^{ix} - e^{-ix}}{2}$, $\frac{d}{dx} \sin x = i\frac{e^{ix}+e^{-ix}}{2}=i\cos x$, not $\cos x$. Why is this?

Answer 1

The equation for $\sin$ should be $$\sin x = \frac{e^{ix} - e^{-ix}}{2\color{red}i}.$$