What is the definition of $e^{ix}$? This might seem silly, but in proving Euler Formula taught in Calculus classes, we make the assumption that
$\frac{d}{dx}e^{ix} = ie^{ix}$
However $e^{g}$ Pre-Euler’s Formula, only takes in real numbers for g.  If we tried to use the Chain Rule where $g = ix$ we have no definition for that.  
My question is, how do we know, Pre-Euler’s Formula, that $e^{z}$ exists for complex values of z, and how do we know that $e^{ix}$ is differentiable at $ix$.  Basically what is the definition of $e^{ix}$ before knowing Euler's Formula.

When I was teaching, I always used the series definition, as explained in @ErikParkinson’s answer. But you may also define
$$
e^z=\lim_{n\to\infty}\left(1+\frac zn\right)^n\,.
$$
When you look at this closely, the formula $e^{it}=\cos t+i\sin t$ becomes very reasonable.
