As in my post about the exponential power series, I set out from the fact that the derivative of the exponential function is the exponential function itself.
The trigonometric functions *sine* and *cosine* are continuous and periodic and if you cycle through their derivatives, they eventually re-appear.
This seems promising.
Maybe they could be used to represent the exponential function?
$$
\begin{align*}
(\cos \theta)' &= -\sin \theta \\
(\sin \theta)' &= \cos \theta
\end{align*}
$$

Take a look at the unit circle.
For any angle \(\theta\), \(\cos \theta\) is the *x*-value and \(\sin \theta\) the *y*-value of a point on this circle.
Now take the derivative of both functions and stay on the same axis such that you get a new point with \(-\sin \theta\) as *x*-value and \(\cos \theta\) as *y*-value.
To get to the original point, all you need is a clockwise rotation of \(\pi / 2\) around the origin.

This idea can be represented nicely in the complex plane, where the *x*-axis represents the real part and the *y*-axis the imaginary part of a complex number.
To get to a point \(z\) on the circle, add the real and imaginary part \((z = \cos \theta + i \sin \theta).\)
Now take the derivative of this sum.
$$
\frac{d}{d \theta} \left( \cos \theta + i \sin \theta \right) = -\sin \theta + i \cos \theta
$$

Multiplication with \(i\) must be a counterclockwise rotation of \(\pi / 2\) around the origin since it takes the point \((1, 0)\) to \((0, i).\) So I multiply both sides by \(1 / i\) to rotate clockwise. $$ \begin{align*} \frac{1}{i} \cdot \frac{d}{d \theta} \left( \cos \theta + i \sin \theta \right) &= \frac{-\sin \theta + i \cos \theta}{i}\\ &= \frac{-\sin \theta}{i} + \cos \theta \\ &= \cos \theta - \frac{\sin \theta (-i)}{i (-i)} \\ &= \cos \theta + i \sin \theta \end{align*} $$

Now I'm looking for a function \(f\) such that the following equation holds. $$ \frac{1}{i} \cdot \frac{d f}{d \theta} = f $$ If you know the chain rule, at this point it becomes quite clear that the function must be \(f(\theta) = e^{i \theta}\) because \(\frac{d}{d \theta} e^{i \theta} = e^{i \theta} i.\) So we end up with Euler's famous formula. $$ e^{i \theta} = \cos \theta + i \sin \theta $$