# Natural Exponential Function eˣ: Graph, Derivative, Integral and e

> The natural exponential function e^x: what e is, its graph, why it is its own derivative and integral, key limits, series and worked examples.

Source: https://practicecalculus.com/functions/natural-exponential-function
Updated: 2026-11-03

$$ f(x) = e^x $$

## At a glance

- **Domain:** $(-\infty, \infty)$
- **Range:** $(0, \infty)$
- **Derivative:** $e^x$
- **Integral:** $e^x + C$
- **Value of e:** $e \approx 2.71828$
- **Inverse:** $\ln x$
- **Asymptote:** $y = 0$ as $x \to -\infty$

## Key takeaways

- The natural exponential function $e^x$ uses the base $e \approx 2.71828$.
- It is the only function (up to a constant multiple) that equals its own derivative: $\frac{d}{dx}e^x = e^x$.
- Its integral is $e^x + C$, and with the chain rule $\frac{d}{dx}e^{g(x)} = e^{g(x)}g'(x)$.
- $e^x$ is always positive, always increasing, and is the inverse of $\ln x$.

The **natural exponential function** $f(x) = e^x$ is the exponential function whose base is Euler's number $e \approx 2.71828$. It is the single most important function in calculus, because it is its own derivative: the rate at which $e^x$ grows always equals its current value.

## Formula and the number e

$$f(x) = e^x, \qquad e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n \approx 2.71828$$

Every [exponential function](/functions/exponential-functions) $b^x$ has a slope at $x = 0$ proportional to its height. The base $e$ is the one choice that makes the slope at $x = 0$ exactly $1$, which is why derivatives of $e^x$ come out so clean.

## Graph of eˣ

```graph
title: 'Graph of the natural exponential function y = eˣ'
caption: 'The curve passes through $(0, 1)$ with slope $1$. The dashed tangent line $y = x + 1$ shows that slope; the curve stays above it everywhere.'
x: [-4, 2.5]
y: [-1, 8]
functions:
  - expr: e^x
    label: y = eˣ
  - expr: x + 1
    label: tangent y = x + 1
    dashed: true
points:
  - [0, 1, '(0, 1)']
  - [1, 2.718, '(1, e)']
```

## Key properties

- **Domain** all reals, **range** $(0, \infty)$: $e^x$ is always positive.
- **Always increasing** and **always concave up**, since $f'(x) = f''(x) = e^x > 0$.
- **Horizontal asymptote** $y = 0$: $\lim_{x \to -\infty}e^x = 0$, while $\lim_{x \to \infty}e^x = \infty$.
- **Inverse of the natural log:** $e^{\ln x} = x$ for $x > 0$ and $\ln(e^x) = x$ for all $x$. See the [natural logarithm](/functions/natural-logarithm).
- **Exponent rules:** $e^{a+b} = e^ae^b$, $e^{a-b} = \frac{e^a}{e^b}$, $(e^a)^b = e^{ab}$, $e^0 = 1$.
- **Grows faster than any power:** $\lim_{x \to \infty}\frac{x^n}{e^x} = 0$ for every $n$.
- **Power series:** $e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$, valid for all $x$.

## Derivative of eˣ

$$\frac{d}{dx}e^x = e^x$$

With the [chain rule](/rules/chain-rule):

$$\frac{d}{dx}e^{g(x)} = e^{g(x)}\,g'(x)$$

The exponent is never brought down in front. That is the [power rule](/rules/power-rule), which does not apply here.

## Integral of eˣ

$$\int e^x\,dx = e^x + C, \qquad \int e^{kx}\,dx = \frac{1}{k}e^{kx} + C \quad (k \ne 0)$$

## Worked examples

**Example 1.** $\frac{d}{dx}e^{3x} = 3e^{3x}$.

**Example 2: chain rule.** $\frac{d}{dx}e^{x^2} = 2x\,e^{x^2}$.

**Example 3: product rule.** $\frac{d}{dx}\left(x e^x\right) = e^x + xe^x = (1 + x)e^x$.

**Example 4: definite integral.** $\int_0^{\ln 2}e^x\,dx = e^{\ln 2} - e^0 = 2 - 1 = 1$.

**Example 5: u-substitution.** With $u = x^2$, $\int 2x\,e^{x^2}\,dx = e^{x^2} + C$.

**Example 6: exponential growth.** The solution of $y' = ky$ with $y(0) = y_0$ is $y = y_0e^{kt}$. This is why $e$ appears in population growth, radioactive decay and continuous compound interest.

## Common mistakes

1. **Using the power rule.** $\frac{d}{dx}e^x \ne xe^{x-1}$.
2. **Forgetting the chain rule factor.** $\frac{d}{dx}e^{5x} = 5e^{5x}$, not $e^{5x}$.
3. **Integrating $e^{kx}$ without dividing by $k$.** $\int e^{4x}\,dx = \frac{1}{4}e^{4x} + C$.
4. **Treating $e^2$ as a variable.** $e^2$ is a constant, so its derivative is $0$.
5. **Writing $e^{a+b} = e^a + e^b$.** Exponents add when you multiply: $e^{a+b} = e^ae^b$.

## How it connects

$e^x$ is the inverse of $\ln x$ and the building block of the [hyperbolic functions](/functions/hyperbolic-functions). Any exponential can be rewritten with base $e$: $b^x = e^{x\ln b}$. For a full walkthrough with more practice, read the guide to [derivatives of exponential and logarithmic functions](/blog/derivatives-of-exponential-and-logarithmic-functions), and see [Taylor series](/blog/taylor-series) for its series expansion.

## Frequently asked questions

### What is the derivative of e^x?

The derivative of $e^x$ is $e^x$. At every point the slope of the graph equals its height, which is the defining property of the base $e$.

### What is the integral of e^x?

$\int e^x\,dx = e^x + C$. More generally, $\int e^{kx}\,dx = \frac{1}{k}e^{kx} + C$ for any constant $k \ne 0$.

### What is the number e?

The number $e \approx 2.71828$ is the limit $\lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n$. Equivalently, it is the unique base for which the exponential $b^x$ has slope exactly $1$ at $x = 0$.

### What is the derivative of e^(2x)?

By the chain rule, $\frac{d}{dx}e^{2x} = e^{2x} \cdot 2 = 2e^{2x}$. Multiply by the derivative of the exponent.

### Can e^x ever be zero or negative?

No. $e^x > 0$ for every real $x$. It approaches $0$ as $x \to -\infty$ but never reaches it, so $y = 0$ is a horizontal asymptote.