# Derivative of e^(x²): Formula, Steps and Examples

> The derivative of e^(x²) is 2x·e^(x²). See the chain rule steps, the second derivative, how it differs from (eˣ)², and examples like e^(−x²).

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

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

## At a glance

- **Derivative:** $2x\,e^{x^2}$
- **Second derivative:** $(4x^2 + 2)\,e^{x^2}$
- **Critical point:** $x = 0$, a minimum
- **Related function:** $\dfrac{d}{dx}e^{-x^2} = -2x\,e^{-x^2}$
- **Integral:** $\int e^{x^2}\,dx$ has no elementary formula

## Key takeaways

- The derivative of $e^{x^2}$ is $2x\,e^{x^2}$: copy the exponential, then multiply by $2x$.
- $e^{x^2}$ and $(e^x)^2 = e^{2x}$ are different functions with different derivatives.
- The second derivative is $(4x^2 + 2)e^{x^2}$, which is always positive, so the graph is always concave up.
- Differentiating $e^{x^2}$ is easy, but $\int e^{x^2}\,dx$ has no formula in elementary functions.

The **derivative of $e^{x^2}$ is $2x\,e^{x^2}$**. The exponential is copied unchanged and multiplied by the derivative of the exponent $x^2$.

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

## Step by step with the chain rule

The function is a composition: an outer exponential $e^u$ and an inside $u = x^2$. By the [chain rule](/rules/chain-rule):

1. Differentiate the outside and keep the inside: since the [derivative of eˣ](/derivatives/e-to-the-x) is $e^x$, this gives $e^{x^2}$.
2. Differentiate the inside: $\frac{d}{dx}x^2 = 2x$.
3. Multiply: $e^{x^2}\cdot 2x = 2x\,e^{x^2}$.

It is usual to write the $2x$ in front so nobody mistakes it for part of the exponent.

## Graph of e^(x²) and its derivative

```graph
title: 'Graph of y = e^(x²) and its derivative y = 2x·e^(x²)'
caption: 'The function has a minimum at $(0, 1)$, where the derivative crosses zero. Away from $0$ both grow very fast.'
x: [-2, 2]
y: [-8, 8]
functions:
  - expr: exp(x^2)
    label: y = e^(x²)
    domain: [-1.4, 1.4]
  - expr: 2x*exp(x^2)
    label: y = 2x·e^(x²) (derivative)
    dashed: true
    domain: [-1.1, 1.1]
points:
  - [0, 1, 'minimum (0, 1)']
```

The derivative $2x\,e^{x^2}$ is negative for $x < 0$ and positive for $x > 0$, because $e^{x^2}$ is always positive. So the graph falls, bottoms out at $x = 0$, then rises. It is symmetric about the y-axis, and its derivative is an odd function.

## Second derivative

Differentiate $2x\,e^{x^2}$ with the [product rule](/rules/product-rule):

$$\frac{d^2}{dx^2}e^{x^2} = 2\cdot e^{x^2} + 2x\cdot 2x\,e^{x^2} = (4x^2 + 2)\,e^{x^2}$$

Since $4x^2 + 2 > 0$, the curve is concave up everywhere and has no inflection points.

## e^(x²) versus (eˣ)²

These look alike but are different functions:

| Function | Simplified | Derivative |
| --- | --- | --- |
| $e^{x^2}$ | $e^{(x^2)}$ | $2x\,e^{x^2}$ |
| $(e^x)^2$ | $e^{2x}$ | $2e^{2x}$ |

At $x = 3$: $e^{x^2} = e^9 \approx 8103$, but $(e^x)^2 = e^6 \approx 403$. See the [derivative of e^(2x)](/derivatives/e-to-the-2x) for the second one.

## Worked examples

**Example 1: the bell curve.** For $f(x) = e^{-x^2}$, the inside derivative is $-2x$:

$$f'(x) = -2x\,e^{-x^2}$$

This shape is the core of the normal distribution. The maximum is at $x = 0$.

**Example 2: a constant in the exponent.** For $g(x) = e^{3x^2 + 1}$,

$$g'(x) = 6x\,e^{3x^2 + 1}$$

**Example 3: a product.** For $h(x) = x\,e^{x^2}$,

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

Notice that $h'(x)$ is always positive, so $h$ is always increasing.

## Common mistakes

1. **Forgetting the $2x$.** $\frac{d}{dx}e^{x^2} = e^{x^2}$ is wrong; that only works for plain $e^x$.
2. **Writing $e^{2x}$.** The exponent never changes when you differentiate an exponential.
3. **Using the power rule.** $x^2 e^{x^2 - 1}$ treats the exponential as a power of $e$ with a variable base. It is not.
4. **Expecting an easy integral.** You can differentiate $e^{x^2}$ in one line, but its antiderivative has no elementary formula. Read the [integral of e^(x²)](/integrals/e-to-the-x-squared) for why, and what to do instead.

Practise more chain rule exponentials with [derivatives of exponential and logarithmic functions](/practice-problems/derivatives-of-exponential-and-logarithmic-functions).

## Frequently asked questions

### What is the derivative of e^(x²)?

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

### What is the second derivative of e^(x²)?

Use the product rule on $2x\,e^{x^2}$: $2e^{x^2} + 2x\cdot 2x\,e^{x^2} = (4x^2 + 2)e^{x^2}$.

### What is the derivative of e^(−x²)?

The inside derivative is $-2x$, so the answer is $-2x\,e^{-x^2}$.

### Is e^(x²) the same as (e^x)²?

No. $(e^x)^2 = e^{2x}$, whose derivative is $2e^{2x}$. In $e^{x^2}$ only the $x$ is squared.