Q:

How did Euler prove $\frac{d}{dx}e^{ix}=ie^{ix}$?

Surely, $\frac{d}{dx}e^{ax}$ is $ae^{ax}$ when $a$ is a real number. However, it does not mean $\frac{d}{dx}e^{ax}$ is $ae^{ax}$ when $a$ is a complex number, I think. if he didn’t prove it, Euler’s formula is not a formula but a definition of $e^{ix}$. Please tell me whether Euler’s formula is a formula or a definition.
Additionally, I want to know whether Euler thought this equation as a formula or a definition when he found this.

A:

He used his own formula, I guess, and applied the usual derivatives of trigonometric functions:
$$\forall\,x\in\Bbb R\;,\;\;e^{ix}=\cos x+i\sin x\implies\left(e^{ix}\right)'=-\sin x+i\cos x=i\left(\cos x+i\sin x\right)=ie^{ix}$$

