# Derivative of eˣ: Formula, Proof and Examples

> The derivative of eˣ is eˣ. See why with a limit proof, the chain rule form for e^u, a graph and worked examples like e^(3x) and x²eˣ.

Source: https://practicecalculus.com/derivatives/e-to-the-x
Updated: 2026-11-06

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

## At a glance

- **Derivative:** $e^x$
- **Chain rule form:** $\dfrac{d}{dx}e^u = e^u\,u'$
- **Every higher derivative:** $e^x$
- **Value of e:** $e \approx 2.71828$
- **Integral version:** $\int e^x\,dx = e^x + C$

## Key takeaways

- The derivative of $e^x$ is $e^x$. It is the only function (up to a constant multiple) that is its own derivative.
- The proof rests on the limit $\lim_{h \to 0}\frac{e^h - 1}{h} = 1$, which is what makes $e$ special.
- With the chain rule, $\frac{d}{dx}e^u = e^u\,u'$: copy the exponential, then multiply by the derivative of the exponent.
- The power rule does not apply to $e^x$, because the variable is in the exponent.

The **derivative of $e^x$ is $e^x$**. The function and its slope are equal at every point, which makes $e^x$ the most important function in calculus.

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

## Graph of eˣ

```graph
title: 'Graph of y = eˣ and its tangent line at x = 0'
caption: 'At every point, the height of $e^x$ equals its slope. At $x = 0$ both are $1$, so the tangent line there is $y = x + 1$.'
x: [-4, 3]
y: [-1, 8]
functions:
  - expr: exp(x)
    label: y = eˣ (and its derivative)
  - expr: x + 1
    label: tangent y = x + 1
    dashed: true
points:
  - [0, 1, '(0, 1), slope 1']
  - [1, 2.718, '(1, e), slope e']
```

Because $e^x > 0$ for every $x$, the slope is always positive, so $e^x$ always increases. And because the slope grows with the height, the curve bends upward faster and faster.

## Proof from the limit definition

Use the [limit definition of the derivative](/blog/limit-definition-of-the-derivative):

$$\frac{d}{dx}e^x = \lim_{h \to 0}\frac{e^{x + h} - e^x}{h} = \lim_{h \to 0}\frac{e^x(e^h - 1)}{h} = e^x\lim_{h \to 0}\frac{e^h - 1}{h}$$

For any base $a$, the derivative of $a^x$ is $a^x$ times the constant $\lim_{h \to 0}\frac{a^h - 1}{h}$. For $a = 2$ that constant is about $0.693$. For $a = 3$ it is about $1.099$. The number $e \approx 2.71828$ is the base where the constant is exactly $1$:

$$\lim_{h \to 0}\frac{e^h - 1}{h} = 1 \quad\Longrightarrow\quad \frac{d}{dx}e^x = e^x$$

You can check this numerically: with $h = 0.001$, $\frac{e^{0.001} - 1}{0.001} \approx 1.0005$.

## Chain rule version: derivative of e^u

When the exponent is something other than $x$, use the [chain rule](/rules/chain-rule):

$$\frac{d}{dx}e^u = e^u \cdot u'$$

Copy the exponential exactly as it is, then multiply by the derivative of the exponent.

| Function | Exponent $u$ | Derivative |
| --- | --- | --- |
| $e^{3x}$ | $3x$ | $3e^{3x}$ |
| $e^{-x}$ | $-x$ | $-e^{-x}$ |
| $e^{x^2}$ | $x^2$ | $2xe^{x^2}$ |
| $e^{\sin x}$ | $\sin x$ | $\cos x\,e^{\sin x}$ |
| $e^{\sqrt{x}}$ | $\sqrt{x}$ | $\frac{e^{\sqrt{x}}}{2\sqrt{x}}$ |

Some of these have their own pages: [e^(2x)](/derivatives/e-to-the-2x), [e^(−x)](/derivatives/e-to-the-negative-x) and [e^(x²)](/derivatives/e-to-the-x-squared).

## Worked examples

**Example 1: product rule.** For $f(x) = x^2e^x$, the [product rule](/rules/product-rule) gives

$$f'(x) = 2xe^x + x^2e^x = e^x(x^2 + 2x)$$

Factoring out $e^x$ makes it easy to find where $f'(x) = 0$: at $x = 0$ and $x = -2$.

**Example 2: quotient rule.** For $g(x) = \frac{e^x}{x}$, the [quotient rule](/rules/quotient-rule) gives

$$g'(x) = \frac{xe^x - e^x}{x^2} = \frac{e^x(x - 1)}{x^2}$$

**Example 3: higher derivatives.** Every derivative of $e^x$ is $e^x$, so $f^{(n)}(x) = e^x$ for every $n$. This is why the Maclaurin series of $e^x$ is $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$.

## Common mistakes

1. **Using the power rule.** $\frac{d}{dx}e^x \ne xe^{x - 1}$. The power rule needs a constant exponent.
2. **Forgetting the chain rule factor.** $\frac{d}{dx}e^{5x} = 5e^{5x}$, not $e^{5x}$.
3. **Changing the exponent.** The derivative of $e^{x^2}$ keeps $e^{x^2}$; it does not become $e^{2x}$.
4. **Treating $e$ as a variable.** $e$ is a constant, so $\frac{d}{dx}e^2 = 0$ and $\frac{d}{dx}x^e = ex^{e - 1}$.

The reverse fact, $\int e^x\,dx = e^x + C$, is on the [integral of eˣ](/integrals/e-to-the-x) page. The inverse function is the natural log; see the [derivative of ln x](/derivatives/ln-x) and the [natural exponential function](/functions/natural-exponential-function).

## Frequently asked questions

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

The derivative of $e^x$ is $e^x$. The function is its own derivative.

### Why is the derivative of e^x equal to e^x?

Because $\frac{d}{dx}e^x = e^x\lim_{h \to 0}\frac{e^h - 1}{h}$, and $e$ is exactly the number that makes that limit equal $1$.

### What is the derivative of e^(3x)?

By the chain rule it is $3e^{3x}$.

### Is the derivative of e^x equal to x·e^(x−1)?

No. That is the power rule, which only works when the exponent is a constant. Here the exponent is the variable, so the answer is $e^x$.