A tangent follows the local direction of a smooth curve at a chosen point. Draw a straight line through that point, judging its direction from the nearby curve on both sides. Then choose two well-separated, readable points on the tangent and calculate its gradient. Example At (2,4) on y=x2, the drawn tangent passes through approximately (1,0) and (3,8). Its gradient is approximately 8-03-1=4. Use points on the tangent, not two arbitrary points on the curve. Use coordinate differences, not the apparent angle on the page: the two axes may have different scales. A tangent can cross a curve or meet it elsewhere; “only one intersection” is not its definition.

A tangent follows the local direction of a smooth curve at a chosen point. Draw a straight line through that point, judging its direction from the nearby curve on both sides. Then choose two well-separated, readable points on the tangent and calculate its gradient.

At \((2,4)\) on \(y=x^2\), the drawn tangent passes through approximately \((1,0)\) and \((3,8)\). Its gradient is approximately \(\dfrac{8-0}{3-1}=4\).

Use points on the tangent, not two arbitrary points on the curve. Use coordinate differences, not the apparent angle on the page: the two axes may have different scales. A tangent can cross a curve or meet it elsewhere; “only one intersection” is not its definition.

Estimating a gradient with a tangent

Estimating a gradient with a tangent