Why Is the Derivative of x² Equal to 2x?

It's the first rule most calculus students learn: the derivative of x² is 2x. Bring the exponent down, subtract one from it. Easy to memorize, and easy to forget why.

But there's a picture that makes it obvious. Take a square with side x, grow each side by a tiny amount, and look at where the extra area goes. The "2" in 2x is two strips, and the reason the formula is so clean is a corner so small it disappears.

The Answer Comes From Throwing Something Away

When you calculate how much a square's area grows, you get three pieces. The derivative keeps two of them and discards the third. That can feel like cheating. For about 150 years after calculus was invented, some critics said it was cheating.

The resolution, the idea of a limit, is what turned calculus from a brilliant tool into rigorous mathematics.

What a Derivative Measures

The derivative of f(x) tells you how fast f changes as x changes. It's the ratio:

change in output / change in input

measured over a vanishingly small change in input. For a straight line, it's just the slope. For a curve like y = x², the slope changes from point to point, and the derivative gives the slope at each point.

The Picture: Growing a Square

A square with side x has area x². Now increase the side by a small amount h. The new area is (x + h)².

Draw it. The new, bigger square is made of:

• The original square: x²
• A thin strip along the right side: x × h
• A thin strip along the top: x × h
• A tiny square in the corner: h × h = h²

(x + h)² = x² + 2xh + h²

So the extra area is 2xh + h². The two strips give the 2x. The corner, h², is tiny: if h = 0.01, the strips contribute 0.02x while the corner is only 0.0001.

The Algebra: The Limit Definition

The derivative is defined as:

f′(x) = lim (h → 0) [f(x + h) − f(x)] / h

For f(x) = x²:

[f(x + h) − f(x)] / h = [(x² + 2xh + h²) − x²] / h
                      = (2xh + h²) / h
                      = 2x + h

As h shrinks toward 0, 2x + h approaches 2x:

f′(x) = 2x

The corner square becomes the "+ h" term, and it vanishes in the limit. See the definition on the calculus formulas page.

Checking With Numbers

At x = 3, the derivative should be 6. Let's compute the average rate of change for smaller and smaller h:

The ratios close in on 6. The leftover amount is always exactly h, the corner's contribution. Compare with the growth of consecutive values on the square numbers list: from 9 to 16 to 25, the gaps are 7 and 9, close to 2x + 1.

The Pattern: The Power Rule

The same picture extends to cubes. A cube with side x has volume x³. Grow each side by h, and the extra volume is mostly three thin slabs of area x² and thickness h, one per face you extend:

(x + h)³ = x³ + 3x²h + 3xh² + h³

Divide by h and let h → 0, and only 3x² survives. In general, the power rule says:

d/dx (xⁿ) = n·xⁿ⁻¹

The n counts how many "faces" grow, and xⁿ⁻¹ is the size of each face. The binomial expansion behind this is on the algebra formulas page.

An Insider Reference: "The Ghosts of Departed Quantities"

Isaac Newton and Gottfried Wilhelm Leibniz developed calculus independently in the second half of the 1600s. Leibniz published first, in 1684. Both relied on "infinitely small" quantities: they divided by h as if it were non-zero, then set it to zero at the end.

In 1734, the philosopher George Berkeley, an Anglican bishop, published The Analyst, a sharp critique. If h isn't zero, you can't drop the h term. If h is zero, you couldn't divide by it. Berkeley mocked these vanishing increments as "the ghosts of departed quantities."

He had a point. The fix came in the 19th century, when Augustin-Louis Cauchy and Karl Weierstrass defined limits precisely using the ε–δ definition. You never set h equal to 0. Instead, you show that 2x + h can be made as close to 2x as anyone demands, by choosing h small enough. The ghosts were finally laid to rest.

What 2x Means on a Graph

On the parabola y = x², the derivative gives the slope of the tangent line:

• At x = 0, the slope is 0: the bottom of the curve is flat
• At x = 1, the slope is 2
• At x = −2, the slope is −4: the curve is falling steeply

The slope doubles as x doubles, which is why the parabola gets steeper and steeper.

Two Concepts Worth Knowing

Limit

A limit describes the value an expression approaches as its input approaches some value, without requiring the input to actually reach it. Limits are the foundation of all of calculus.

Tangent Line

A tangent line touches a curve at a point and has the same slope as the curve there. The derivative is the slope of the tangent line.

Quick Answer: Why Is the Derivative of x² Equal to 2x?

Increase x by a small amount h. Then x² becomes x² + 2xh + h², so the change divided by h is 2x + h. As h approaches 0, that approaches 2x. Geometrically, growing a square adds two thin strips of area x·h and a tiny corner h² that becomes negligible.

Try Them Yourself

• Calculus Formulas: the limit definition and power rule
• Algebra Formulas: binomial expansions
• Square Numbers List: see how squares grow
• Cube Numbers List: check the 3x² rule numerically
• Scientific Calculator: compute (x + h)² − x² for tiny h
• The Mathematics Behind Machine Learning: where derivatives power modern AI

Draw a 5 × 5 square on graph paper, then extend it to 6 × 6. Count the new cells: two strips of 5 and one corner cell, 11 in all. That's 2x + h with x = 5 and h = 1.