Definition:Exponential Function/Real/Differential Equation

Definition
Let $\exp: \R \to \R_{>0}$ denote the (real) exponential function. The exponential function can be defined as the unique solution $y = f(x)$ to the first order ODE:


 * $\dfrac{\mathrm d y}{\mathrm d x} = y$

...satisfying the initial condition $f(0) = 1$.

That is, the defining property of $\exp$ is that it is its own derivative. The number $\exp x$ is called the exponential of $x$.