Let $c$ be a constant and $f:\mathbb R\to\mathbb R$ be a differentiable function with the first-order ODE.
$f'(x)=cf(x)$ for all $x\in\mathbb R$.
Let $A:=f(0).$ Then the exponential function.
$f(x)=A\exp(cx)$
is the solution of the above first-order ODE.
In particular, the exponential function $\exp:\mathbb R\to\mathbb R$ is the differential function $f,$ which is uniquely defined by $f'=f$ and $f(0)=1$.
Proofs: 1