# Derivative of e^(2x): Formula, Steps and Examples

> The derivative of e^(2x) is 2e^(2x). See the chain rule steps, a second proof using (eˣ)², higher derivatives, a graph and worked examples.

Source: https://practicecalculus.com/derivatives/e-to-the-2x
Updated: 2026-11-06

$$ \frac{d}{dx}\,e^{2x} = 2e^{2x} $$

## At a glance

- **Derivative:** $2e^{2x}$
- **Second derivative:** $4e^{2x}$
- **nth derivative:** $2^n e^{2x}$
- **General form:** $\dfrac{d}{dx}e^{kx} = ke^{kx}$
- **Integral version:** $\int e^{2x}\,dx = \frac{1}{2}e^{2x} + C$

## Key takeaways

- The derivative of $e^{2x}$ is $2e^{2x}$: the exponent's derivative, $2$, comes out front.
- More generally, $\frac{d}{dx}e^{kx} = ke^{kx}$ for any constant $k$.
- Each extra derivative multiplies by another $2$, so the $n$th derivative is $2^n e^{2x}$.
- Going backwards, the integral divides by $2$: $\int e^{2x}\,dx = \frac{1}{2}e^{2x} + C$.

The **derivative of $e^{2x}$ is $2e^{2x}$**. The exponential stays exactly the same, and the derivative of the exponent, $2$, comes out in front.

$$\frac{d}{dx}\,e^{2x} = 2e^{2x}$$

## Step by step with the chain rule

Write $e^{2x}$ as an outer function $e^u$ with inside $u = 2x$. The [chain rule](/rules/chain-rule) says

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

1. The outer derivative: the [derivative of eˣ](/derivatives/e-to-the-x) is itself, so $e^u$ stays as $e^{2x}$.
2. The inside derivative: $\frac{d}{dx}(2x) = 2$.
3. Multiply: $e^{2x}\cdot 2 = 2e^{2x}$.

The same steps give the general rule $\frac{d}{dx}e^{kx} = ke^{kx}$. For example, $\frac{d}{dx}e^{5x} = 5e^{5x}$ and $\frac{d}{dx}e^{x/3} = \frac{1}{3}e^{x/3}$.

## A second proof without the chain rule

Since $e^{2x} = (e^x)^2 = e^x\cdot e^x$, the [product rule](/rules/product-rule) works too:

$$\frac{d}{dx}(e^x\cdot e^x) = e^x\cdot e^x + e^x\cdot e^x = 2e^{2x}$$

Both methods agree, which is a nice check.

## Graph of e^(2x) and its derivative

```graph
title: 'Graph of y = e^(2x) and its derivative y = 2e^(2x)'
caption: 'The derivative is always exactly twice the height of the function. At $x = 0$ the function is $1$ and the slope is $2$.'
x: [-3, 1.5]
y: [-1, 10]
functions:
  - expr: exp(2x)
    label: y = e^(2x)
  - expr: 2exp(2x)
    label: y = 2e^(2x) (derivative)
    dashed: true
points:
  - [0, 1, '(0, 1)']
  - [0, 2, 'slope 2']
```

The tangent line at $x = 0$ has slope $2$ and passes through $(0, 1)$, so it is $y = 2x + 1$.

## Higher derivatives

Each time you differentiate, another factor of $2$ comes out:

| Derivative | Result |
| --- | --- |
| $f'(x)$ | $2e^{2x}$ |
| $f''(x)$ | $4e^{2x}$ |
| $f'''(x)$ | $8e^{2x}$ |
| $f^{(n)}(x)$ | $2^n e^{2x}$ |

This pattern is why $y = e^{2x}$ solves the differential equation $y'' = 4y$. It also explains the first derivative on its own: $y' = 2y$ says the function grows at a rate equal to twice its current size, which is the rule behind exponential growth with rate constant $2$.

## Worked examples

**Example 1: product rule.** For $f(x) = xe^{2x}$,

$$f'(x) = 1\cdot e^{2x} + x\cdot 2e^{2x} = e^{2x}(1 + 2x)$$

Since $e^{2x} > 0$, the only critical point is where $1 + 2x = 0$, at $x = -\frac{1}{2}$.

**Example 2: a shifted exponent.** For $g(x) = e^{2x + 1}$, the inside derivative is still $2$:

$$g'(x) = 2e^{2x + 1}$$

**Example 3: a constant multiple.** For $h(x) = 5e^{2x} - e^{-2x}$,

$$h'(x) = 10e^{2x} + 2e^{-2x}$$

The second term flips sign because $\frac{d}{dx}(-2x) = -2$.

## Common mistakes

1. **Forgetting the 2.** $\frac{d}{dx}e^{2x} = e^{2x}$ is the most common slip. Always multiply by the inside derivative.
2. **Bringing the exponent down like a power.** $2x\,e^{2x - 1}$ is wrong; the power rule does not apply.
3. **Writing $2e^{x}$.** The exponent does not change. It stays $2x$.
4. **Mixing up the derivative and the integral.** The derivative multiplies by $2$; the integral divides by $2$.

For the reverse direction, see the [integral of e^(2x)](/integrals/e-to-the-2x). For more practice, try [derivatives of exponential and logarithmic functions](/practice-problems/derivatives-of-exponential-and-logarithmic-functions).

## Frequently asked questions

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

The derivative of $e^{2x}$ is $2e^{2x}$, by the chain rule.

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

Differentiate $2e^{2x}$ once more to get $4e^{2x}$.

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

By the product rule, $e^{2x} + 2xe^{2x} = e^{2x}(1 + 2x)$.

### Is the derivative of e^(2x) equal to 2x·e^(2x − 1)?

No. That treats $e^{2x}$ like a power of $x$. The exponent contains the variable, so use the exponential rule with the chain rule.