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Why the Derivative of x² is 2x

Discover why the derivative of x² is 2x by exploring the changing area of a growing square and calculating the slope of the function on a graph.

ddx(x2)=?\frac{d}{dx}(x^2) = ?
You might have memorised the rule that the derivative of x squared is 2x. But why is it exactly 2x? Let's build some intuition.
Step-by-step solver
1

Understand the definition of a derivative

The derivative measures the instantaneous rate of change of a function. For a function f(x), it is the limit of (f(x+h) - f(x)) / h as h approaches 0.

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
2

Set up the limit for x^2

Substitute f(x) = x^2 into the definition.

f′(x)=lim⁡h→0(x+h)2−x2hf'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}
3

Expand the bracket

Multiply out (x+h)^2.

f′(x)=lim⁡h→0x2+2xh+h2−x2hf'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}
4

Simplify the expression

The x^2 terms cancel out.

f′(x)=lim⁡h→02xh+h2hf'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h}
5

Factor and cancel h

Factor out h from the numerator and cancel it with the h in the denominator (since h is not exactly zero yet).

f′(x)=lim⁡h→0(2x+h)f'(x) = \lim_{h \to 0} (2x + h)
6

Evaluate the limit

As h gets closer and closer to 0, the expression becomes just 2x.

f′(x)=2xf'(x) = 2x

Original question

Explain why the derivative of x squared is 2x, using a graph.

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