Gradient of the Curve at a Point (Tangent)

A tangent at a point on a smooth curve is the straight line through that point with the same local direction as the curve.

• Gradient of the curve at a point is defined as the gradient of the tangent to the curve at that point.

A curve can have different gradients at different points. A straight line has a constant gradient.

In these drawings, compare the local direction of each candidate line with the curve at the stated point.

The correct tangent drawn is shown in Figure 3.

Correction to the image captions: a tangent may cross a curve at its point of contact, or meet the curve elsewhere. The test is whether the line has the same local direction, not whether there is exactly one intersection.