The Origin of the Exponential Function

A problem involving derivatives

We are looking for a function f, defined and differentiable on R, such that:

f' = f and f(0) = 1

This equation states that f is its own derivative. It is accepted that there is a unique function satisfying this property: this is called the exponential function, denoted by exp. We set e = exp(1), an irrational number whose approximate value is e ≈ 2.718.

Exponential notation

We often write exp(x) = e^x. This notation is justified by the algebraic properties of exp, which resemble the rules for powers: e^0 = 1, e^1 = e, e^(a+b) = e^a * e^b.

First values

x	-1	0	1	2
e^x	1/e ≈ 0.37	1	e ≈ 2.72	e^2 ≈ 7.39

Sign and direction of variation

Since f' = f and f(0) = 1 > 0, we can show that f(x) > 0 for all real numbers x (a differentiable function that vanishes would change sign, which would contradict f' = f). Furthermore, f' = f > 0, so f is strictly increasing on ℝ.

Common pitfall

Do not confuse e^x (which is always strictly positive) with x² (which is positive but can be zero). The exponential function NEVER equals zero: the equation e^x = 0 has no solution, regardless of the value of x.

Key points

• exp is the only function such that exp' = exp and exp(0) = 1.
• e^x > 0 for all real x.
• exp is strictly increasing on R.