First, we start with those equations for sine and cosine with an infinite number of terms:
\begin{align}
\sin x &= \left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - \cdots \right) \\[8pt]
\cos x &= \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \cdots \right) \\
\end{align}
We also take a formula for the exponential function also with an infinite number of terms:
$$e^{x} = \sum_{k = 0}^{\infty} \frac{x^k}{k!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$$
Those series can be proved independently of the Euler's theorem, without relying into arbitrarily-looking definitions.
So, this proceeds to:
$$e^{ix} = 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \frac{(ix)^6}{6!} + \frac{(ix)^7}{7!} + \frac{(ix)^8}{8!} + \cdots $$
Now, considering that $i^2=-1$, that $i^3=-i$ and that $i^4=1$, this can be rewritten as:
$$e^{ix} = 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} - \frac{x^6}{6!} - \frac{ix^7}{7!} + \frac{x^8}{8!} + \cdots$$
Then, we split it into two sums, one without the $i$s and one with them:
$$e^{ix} = \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \cdots \right) + i\left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \right)$$
Replacing those power series with the sine and cosine power series:
$$e^{ix} = \cos x + i\sin x$$
Finally, by setting $x := \pi$:
$$\begin{align}
e^{i\pi} &= \cos \pi + i\sin \pi\\
e^{i\pi} &= -1 + 0i\\
e^{i\pi} &= -1\\
\end{align}$$
BTW, this is exactly the power series formula.
The formula $i = \sqrt{-1}$ is an arbitrary definition. Someone stated that because it would be "cool". This is something different from the case of $e^{ix} = \cos x + i\sin x$, which is not a simple arbitrary definition and can be obtained mathematically from more fundamental calculations. Defining something is very different than obtaining or inferring something.
This shows that that formula can be obtained without needing to resort to integral calculus or to derivatives nor to use some previously made-up definition of how to extend the exponential function into the $\mathbb{C}$ realm (in fact, this is exactly what this formula tells without being something made-up). All you need to have to reach it is to have a power-series formula for sine, cosine and exponential, all in the $\mathbb{R}$ domain. And as I said before, those power-series can be proved independently.