What Is a Derivative?

An average rate of change describes what happened over an interval — a car's average speed over an hour, say. But how fast is something changing at a single instant? That question is what the derivative answers.

From secant to tangent

Start with the difference quotient, (f(x + h) − f(x)) / h — the gradient of the line joining two points on a curve, a distance h apart. Now imagine h shrinking toward zero. The two points slide together, and that line pivots until it becomes the tangent — the line that just touches the curve at a single point.

The definition

The derivative of f at x is the limit of the difference quotient as h → 0:

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

Where this limit exists, it gives the instantaneous rate of change of f at that point — equivalently, the gradient of the tangent line there.

Working it out for f(x) = x²

The definition looks abstract until you apply it. Take f(x) = x².

Step 1 — write out f(x + h):

f(x + h) = (x + h)² = x² + 2xh + h²

Step 2 — subtract f(x):

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

Step 3 — divide by h:

(2xh + h²) / h = 2x + h

Step 4 — let h → 0:

2x + h → 2x

So f'(x) = 2x. That's the general rule for squaring functions: the gradient of f(x) = x² at any point x is simply 2x.

Notation

The same idea is written a few different ways: f'(x), dy/dx, d/dx[f(x)] — all meaning the instantaneous rate of change.

What the sign tells you

A positive derivative means the function is increasing at that point; negative means decreasing; zero means the curve is momentarily flat — often a peak, trough, or point of inflection.

In short

• The derivative is the limit of the difference quotient as h → 0.
• It equals the instantaneous rate of change — the gradient of the tangent line.
• For f(x) = x², applying the definition directly gives f'(x) = 2x.

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