Derivative of the complex exponential

Statement

For each $z \in \mathbb{C}$, the complex exponential is complex differentiable at $z$, and $$\exp'(z) = \exp z.$$

Remarks

Proof: differentiate the series $\sum_{n \ge 0} z^n / n!$ term by term. The derivative of $z^n / n!$ is $z^{n-1} / (n-1)!$ for $n \ge 1$, so the series of derivatives is the same series. With the chain rule, the derivative of $e^{a z}$ is $a e^{a z}$ for each constant $a \in \mathbb{C}$. It follows that $\exp$ is entire.

Explanations

L1 human

As in the real case, the exponential function is its own derivative. You can see it in the power series: differentiating each term gives the term before it, so the whole series does not change.