What is important to know about Rates of change?

Rate of change = gradient of a graph = change in y ÷ change in x.

What is important to know about Rates of change?

On a distance-time graph, gradient = speed.

What is important to know about Rates of change?

On a velocity-time graph, gradient = acceleration.

⚖️

Ratio & Proportion · Higher

Rates of change

Rate of change describes how one quantity changes with respect to another. On a graph, the rate of change is found from the gradient. At Higher tier this includes average and instantaneous rates of change.

Revising rates of change the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Rates of change GCSE” on the site for the complete guide.

🔑

Key facts to remember

1Rate of change = gradient of a graph = change in y ÷ change in x.
2On a distance-time graph, gradient = speed.
3On a velocity-time graph, gradient = acceleration.
4Average rate of change: gradient of the chord between two points.
5Instantaneous rate of change: gradient of the tangent at a point (drawn by hand at GCSE).
6A steeper gradient means a faster rate of change.

📐

Formulas

Gradient (rate of change)

gradient = (y₂ − y₁) / (x₂ − x₁)

✍️

Worked examples

Example 1

A graph shows distance (km) against time (hours). Between t = 1 and t = 4, the distance goes from 20 km to 80 km. Find the average speed.

Working

Average rate of change = gradient of chord
Gradient = (80 − 20) / (4 − 1) = 60 / 3 = 20 km/h

Answer20 km/h

Example 2

On a curved graph, a tangent is drawn at t = 3. The tangent passes through (1, 10) and (5, 30). Find the instantaneous rate of change at t = 3.

Working

Gradient of tangent = (30 − 10) / (5 − 1) = 20 / 4 = 5

AnswerInstantaneous rate of change = 5 units per unit time

⚠️

Common mistakes

✗Calculating the gradient using horizontal ÷ vertical instead of vertical ÷ horizontal.

✗Confusing average rate of change (chord) with instantaneous rate of change (tangent).

✗Not including units in the rate of change (e.g. km/h, m/s).

🎯

Exam tips

✓For average rate of change, calculate the gradient between the two given points.

✓For instantaneous rate of change, draw a tangent at the point and calculate its gradient using two points on the tangent (not the curve).

Ready to test yourself on Rates of change?

Get AI-marked practice questions on exactly this subtopic.

Practice this topic →

← All topics Dashboard

▶️ Watch on YouTube

Free video lessons

Click a topic to search

▶rates of change GCSE Higher maths ▶gradient rate of change GCSE ▶tangent gradient curved graph GCSE Higher ▶average instantaneous rate of change GCSE

Opens YouTube — pick any free GCSE video.