Among all possible bases for exponential functions, one stands out: the number $e \approx 2.71828$. What makes $e$ special isn't its numerical value—it's that $e^x$ is the only exponential function that is its own derivative.
This property makes $e^x$ the natural choice for modeling continuous growth and decay. Whenever something grows "at a rate proportional to its current size," the mathematics naturally leads to $e$.
The constant $e$ appears everywhere: compound interest, population models, probability distributions, physics, and engineering. Understanding why $e$ is special (not just memorizing its value) is key to understanding exponential calculus.
| Property | Value/Formula |
|---|---|
| Approximate Value | $e \approx 2.71828182845904...$ |
| Defining Property | $\displaystyle\lim_{h \to 0} \frac{e^h - 1}{h} = 1$ |
| Key Derivative | $\frac{d}{dx}[e^x] = e^x$ |
| Characterization | $e$ is between 2 and 3, and $e^x$ has slope 1 at $x = 0$ |
| Natural Exponential | $f(x) = e^x$ is called the natural exponential function |
When we differentiate $f(x) = b^x$ using the limit definition:
$$f'(x) = \lim_{h \to 0} \frac{b^{x+h} - b^x}{h} = \lim_{h \to 0} \frac{b^x \cdot b^h - b^x}{h} = b^x \cdot \lim_{h \to 0} \frac{b^h - 1}{h}$$
The derivative of any exponential function equals the function times a constant:
$$\boxed{\frac{d}{dx}[b^x] = b^x \cdot f'(0) \quad \text{where } f'(0) = \lim_{h \to 0} \frac{b^h - 1}{h}}$$
This constant $f'(0)$ depends on the base $b$:
| Base | $\displaystyle\lim_{h \to 0} \frac{b^h - 1}{h}$ |
|---|---|
| $b = 2$ | $\approx 0.693$ |
| $b = 3$ | $\approx 1.099$ |
| $b = ?$ | $= 1$ (this is $e$!) |
$$\boxed{e \text{ is the unique number such that } \lim_{h \to 0} \frac{e^h - 1}{h} = 1}$$
Geometric meaning: The function $f(x) = e^x$ has a tangent line with slope exactly 1 at the point $(0, 1)$.
y
| /
| / y = e^x
| /
| /
| / slope = 1 at (0,1)
| /___
| / tangent line
|/
---+------------------------ x
|(0,1)Numerical evidence:
Since the limit increases with $b$, there must be a unique $b$ between 2 and 3 where the limit equals exactly 1. This number is $e$.
| Decimal Places | Value |
|---|---|
| 2 | 2.72 |
| 5 | 2.71828 |
| 10 | 2.7182818285 |
| 20 | 2.71828182845904523536 |
Important: $e$ is irrational—its decimal expansion never terminates or repeats.
$$\boxed{f(x) = e^x \text{ is the \textbf{natural exponential function}}}$$
Properties of $e^x$:
The equation $\frac{dy}{dx} = y$ (rate of change equals current value) has solution $y = Ce^x$.
Real-world meaning: When something grows at a rate proportional to itself (bacteria, investments, rumors), the mathematics naturally produces $e$.
The number $e$ can be defined in several equivalent ways:
$$\boxed{e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n}$$
This limit arises from compound interest: if you invest $1 at 100% annual interest compounded $n$ times per year, your balance after one year approaches $e$ as $n \to \infty$.
| $n$ | $\left(1 + \frac{1}{n}\right)^n$ |
|---|---|
| 1 | 2 |
| 10 | 2.5937... |
| 100 | 2.7048... |
| 1000 | 2.7169... |
| $\to \infty$ | $e = 2.71828...$ |
| Mistake | Correct Understanding |
|---|---|
| Thinking $e$ is rational | $e$ is irrational (decimal never repeats) |
| Confusing $e$ with $e^1$ | These are the same: $e = e^1 \approx 2.718$ |
| Thinking $e^0 = e$ | $e^0 = 1$ (like any base) |
| Using $e \approx 3$ | Use $e \approx 2.718$ for estimates |
| Thinking other bases work | Only $e^x$ equals its own derivative |
Evaluate (leave answers in exact form or round to 3 decimal places): (a) $e^0$ (b) $e^1$ (c) $e^{-1}$
Compute $\left(1 + \frac{1}{100}\right)^{100}$ to 4 decimal places and compare to $e$.
Verify numerically that $\lim_{h \to 0} \frac{2^h - 1}{h} \neq 1$ by computing the expression for $h = 0.01$ and $h = 0.001$.
Find the equation of the tangent line to $y = e^x$ at the point where $x = 0$.
Show that $\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x$ for any fixed real number $x$.
Hint: Let $m = n/x$ so that $n = mx$, and rewrite the expression.
Question 1: The defining property of $e$ is:
(A) $e \approx 2.718$ (B) $e^1 = e$ (C) The tangent line to $y = e^x$ at $(0, 1)$ has slope 1 (D) $e$ is irrational
(C) While all statements are true, the defining property of $e$ is that $\lim_{h \to 0} \frac{e^h - 1}{h} = 1$, which geometrically means the tangent line at $(0, 1)$ has slope 1. Options (A), (B), and (D) are consequences, not definitions.
Question 2: If $f(x) = 2^x$, then $f'(x)$ equals:
(A) $2^x$ (B) $x \cdot 2^{x-1}$ (C) $2^x \ln 2$ (D) $e^x$
(C) The derivative of $b^x$ is $b^x \ln b$. For base 2, this gives $2^x \ln 2$. Only $e^x$ equals its own derivative; for any other base, there's an extra factor.
Question 3: As $n$ increases, $\left(1 + \frac{1}{n}\right)^n$:
(A) Increases without bound (B) Approaches 1 (C) Approaches $e$ (D) Oscillates
(C) This is one of the fundamental definitions of $e$. The sequence increases but is bounded above by 3, and converges to $e \approx 2.718$.
The "Just Right" Base:
Imagine trying to find an exponential function where the growth rate at $x = 0$ is exactly 100% of the current value.
This "Goldilocks property" is why $e$ appears throughout calculus and nature—it's the base where growth rate perfectly matches current size.
Continuous compounding analogy: If a bank offers 100% interest but compounds it continuously (infinitely often), $1 becomes $e \approx $2.72 after one year.
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|---|---|---|
| Exponential Properties | Skills Index | Derivatives of Exponentials |
Last updated: 2026-01-23