Proposition: Differential Equation of the Exponential Function

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.


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

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983