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The Gradient of a Curve

A curve's gradient at one exact point is found by looking at what happens as two points get closer together.

The Problem With Curves

A straight line has one gradient everywhere. A curve does not — its steepness changes as you move along it. So "the gradient of a curve" only makes sense at a single point . A line joining two points on a curve is called a secant line . A line that touches the curve at just one point, matching its steepness there, is called a tangent line . The gradient of the curve at a point is the gradient of its tangent line at that point.

Change a value and watch:

Make h smaller and smaller. Watch the secant line rotate until it lands on the tangent line.

As the second point slides closer to P, the secant line's gradient gets closer and closer to one fixed number — the gradient of the curve at P.

Example 1 — secant gradients getting closer

f(x) = x^{2} . Find the gradient of the secant line from x = 2 to x = 2 + h, for h = 1, then h = 0.1 . For h = 1 : points are (2, 4) and (3, 9). Gradient = (9 - 4) \div (3 - 2) = 5 For h = 0.1 : points are (2, 4) and (2.1, 4.41). Gradient = (4.41 - 4) \div 0.1 = 4.1 As h gets smaller, the gradient is getting closer to 4 .

Secant gradient of y = x² at x = 2, as h shrinks
Secant gradient of y = x² at x = 2, as h shrinks

The secant gradient is heading straight for 4 . That is the true gradient of y = x^{2} at x = 2 .

Practice

f(x) = x^{2} . Find the secant gradient between x = 3 and x = 3.1 .

Your answer

More lessons in Calculus · Next: Limits · Previous: Rate of Change

The Problem With Curves

A straight line has one gradient everywhere. A curve does not — its steepness changes as you move along it. So "the gradient of a curve" only makes sense at a single point . A line joining two points on a curve is called a secant line . A line that touches the curve at just one point, matching its steepness there, is called a tangent line . The gradient of the curve at a point is the gradient of its tangent line at that point.

Change a value and watch:

Make h smaller and smaller. Watch the secant line rotate until it lands on the tangent line.

As the second point slides closer to P, the secant line's gradient gets closer and closer to one fixed number — the gradient of the curve at P.

Example 1 — secant gradients getting closer

f(x) = x^{2} . Find the gradient of the secant line from x = 2 to x = 2 + h, for h = 1, then h = 0.1 . For h = 1 : points are (2, 4) and (3, 9). Gradient = (9 - 4) \div (3 - 2) = 5 For h = 0.1 : points are (2, 4) and (2.1, 4.41). Gradient = (4.41 - 4) \div 0.1 = 4.1 As h gets smaller, the gradient is getting closer to 4 .

Secant gradient of y = x² at x = 2, as h shrinks
Secant gradient of y = x² at x = 2, as h shrinks

The secant gradient is heading straight for 4 . That is the true gradient of y = x^{2} at x = 2 .

Practice

f(x) = x^{2} . Find the secant gradient between x = 3 and x = 3.1 .

Your answer