How to find the gradient of a curve

A straight line has the same gradient everywhere. A curve doesn't: it gets steeper or shallower as you move along it, so its gradient depends on where you look. That gradient is the rate of change at that point, such as speed on a distance-time graph.

Here x and y are the usual coordinates. At (3, 9) the curve is rising steeply, and its tangent there is the line y = 6x − 9, with gradient 6.

Method 1: draw a tangent (GCSE)

1. Mark the point on the curve where you want the gradient.
2. Draw the tangent with a ruler: a straight line that touches the curve at that point without cutting across it, matching the curve's direction there.
3. Pick two points on the tangent that are far apart and easy to read.
4. Calculate gradient = change in y ÷ change in x.
5. Check the sign: uphill from left to right is positive, downhill is negative.

Example: on a distance-time graph, a tangent drawn at t = 4 s passes through (2, 5) and (6, 13). Gradient = (13 − 5) ÷ (6 − 2) = 8 ÷ 4 = 2, so the speed at 4 seconds is 2 m/s. Because the tangent is drawn by eye, answers within a small range usually get the marks. In GCSE Maths, estimating gradients of curved graphs and interpreting them is Higher tier content, part of A15 in the AQA specification.

Method 2: chords that close in

A chord joins two points on the curve. Its gradient is the average rate of change between them. Move the second point closer and closer to the first, and the chord's gradient homes in on the tangent's. For y = x² at x = 3, with a second point h further along:

The chord gradients approach 6. This idea, letting h shrink towards zero, is called differentiation from first principles, and it is the definition of the derivative:

Try differentiating from first principles yourself, then press Mark to check it in our Calculus course.

Method 3: differentiate (A-level)

Differentiation gives a formula for the gradient everywhere at once. For powers of x, the rule is: multiply by the power, then reduce the power by 1.

Differentiate each term separately, and constants disappear because they don't change. Then substitute the x value of your point.

Example: a cubic

Find the gradient of y = x³ − 4x + 1 at x = 2. Differentiate: dy/dx = 3x² − 4. At x = 2: 3 × 2² − 4 = 12 − 4 = 8. The Derivative Calculator shows each rule as it differentiates, and the Rate of Change Calculator compares average and instantaneous rates.

Gradient zero: turning points

Where a curve's gradient is 0, the tangent is flat: the curve has a turning point, a maximum or minimum. For y = x², dy/dx = 2x = 0 when x = 0, the bottom of the parabola. Setting the gradient equal to zero is how calculus finds maximum and minimum values.

Common mistakes

• Drawing a chord instead of a tangent. A tangent touches the curve at one point. A chord cuts it at two.
• Reading points too close together on a hand-drawn tangent. Far-apart points give a more accurate gradient.
• Forgetting the sign. A curve going down from left to right has a negative gradient.
• Substituting before differentiating. Differentiate first, then put in the x value.
• Mixing up the y value and the gradient. At x = 3 on y = x², y is 9 but the gradient is 6.

Read next

Maths · CalculusParametric differentiation: dy/dx, tangents and normalsHow to differentiate parametric equations: dy/dx = (dy/dt) ÷ (dx/dt), with worked examples for gradients, tangents, normals and stationary points.