# Derivative of x²: Formula, Proof and Examples

> The derivative of x² is 2x. See the limit definition proof, the chain rule version for u², a graph, tangent lines and worked examples like (3x + 1)².

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

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

## At a glance

- **Derivative:** $2x$ for every real $x$
- **Chain rule form:** $\dfrac{d}{dx}\,u^2 = 2u\,u'$
- **Second derivative:** $\dfrac{d^2}{dx^2}\,x^2 = 2$
- **Slope at x = 0:** $0$, the bottom of the parabola
- **Integral version:** $\int x^2\,dx = \frac{x^3}{3} + C$

## Key takeaways

- The derivative of $x^2$ is $2x$: bring the 2 down and lower the exponent to 1.
- The slope of $y = x^2$ at $x = a$ is $2a$, so it is negative on the left, zero at the vertex and positive on the right.
- With the chain rule, $\frac{d}{dx}\,u^2 = 2u\,u'$. Do not forget the $u'$.
- Do not confuse $x^2$ with $2^x$: $\frac{d}{dx}\,2^x = 2^x\ln 2$.

The **derivative of $x^2$ is $2x$**. This is usually the first derivative students compute by hand, and it shows the whole idea of a derivative in one line of algebra.

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

## Graph of x² and its derivative

```graph
title: 'Graph of y = x² and its derivative y = 2x'
caption: 'Where the parabola falls, $2x$ is negative. At the vertex the slope is $0$. Where it rises, $2x$ is positive.'
x: [-3, 3]
y: [-5, 7]
functions:
  - expr: x^2
    label: y = x²
  - expr: 2x
    label: y = 2x (derivative)
    dashed: true
points:
  - [0, 0, 'vertex, slope 0']
  - [1, 1, '(1, 1), slope 2']
```

The derivative grows as $x$ grows, so the parabola gets steeper the further you move from the vertex. That is what concave up looks like.

## Proof from the limit definition

Start from

$$f'(x) = \lim_{h \to 0}\frac{(x + h)^2 - x^2}{h}$$

Expand the square: $(x + h)^2 = x^2 + 2xh + h^2$. The $x^2$ terms cancel:

$$f'(x) = \lim_{h \to 0}\frac{2xh + h^2}{h} = \lim_{h \to 0}\,(2x + h) = 2x$$

You can also get it from the [power rule](/rules/power-rule), or from the [product rule](/rules/product-rule) with $x \cdot x$: $1\cdot x + x \cdot 1 = 2x$.

## The chain rule version: derivative of u²

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

$$\frac{d}{dx}\,u^2 = 2u \cdot u'$$

| Function | Inside $u$ | Derivative $2u\,u'$ |
| --- | --- | --- |
| $(3x + 1)^2$ | $3x + 1$ | $6(3x + 1)$ |
| $(x^2 + 5)^2$ | $x^2 + 5$ | $4x(x^2 + 5)$ |
| $\sin^2 x$ | $\sin x$ | $2\sin x\cos x$ |
| $(\ln x)^2$ | $\ln x$ | $\frac{2\ln x}{x}$ |

The third row is worked out fully on the [derivative of sin²x](/derivatives/sin-squared-x) page.

## Worked examples

**Example 1: a polynomial.** Differentiate $f(x) = 4x^2 - 3x + 8$:

$$f'(x) = 8x - 3$$

**Example 2: a tangent line.** Find the tangent to $y = x^2$ at $x = 3$. The point is $(3, 9)$ and the slope is $2(3) = 6$:

$$y - 9 = 6(x - 3) \quad\Longrightarrow\quad y = 6x - 9$$

**Example 3: where is the slope 5?** Solve $2x = 5$ to get $x = \frac{5}{2}$. The point on the parabola is $\left(\frac{5}{2}, \frac{25}{4}\right)$.

**Example 4: second derivative.** $\frac{d}{dx}(2x) = 2$, so $\frac{d^2}{dx^2}x^2 = 2$. A positive constant second derivative means the parabola bends upward everywhere.

## Common mistakes

1. **Writing $2x^2$.** The exponent must drop by one: $2x^1 = 2x$.
2. **Mixing up $x^2$ and $2^x$.** They look alike but $2^x$ is an exponential. Its derivative is $2^x \ln 2$, see the [derivative of 2ˣ](/derivatives/2-to-the-x).
3. **Dropping $u'$.** $\frac{d}{dx}(5x - 2)^2 = 10(5x - 2)$, not $2(5x - 2)$.
4. **Squaring the derivative.** $\frac{d}{dx}(x^2)$ is not $\left(\frac{d}{dx}x\right)^2 = 1$.

Going backward, $\int x^2\,dx = \frac{x^3}{3} + C$ is on the [integral of x²](/integrals/x-squared) page. Practice more with the [power rule problems](/practice-problems/power-rule).

## Frequently asked questions

### What is the derivative of x²?

The derivative of $x^2$ is $2x$. By the power rule, $\frac{d}{dx}x^2 = 2x^{2-1} = 2x$.

### How do you find the derivative of x² using the limit definition?

Expand $\frac{(x+h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$. As $h \to 0$ this tends to $2x$.

### What is the derivative of 3x²?

It is $6x$. Multiply the derivative of $x^2$ by the constant: $3 \cdot 2x = 6x$.

### What is the derivative of (2x + 1)²?

By the chain rule it is $2(2x + 1) \cdot 2 = 4(2x + 1) = 8x + 4$. Expanding to $4x^2 + 4x + 1$ first gives the same answer.