The gradient of a curve at a point is the gradient of the tangent drawn at that point. Draw a straight line that just touches the curve at the given point without cutting through it, then read off two points on that tangent and apply the gradient formula (y₂ − y₁) ÷ (x₂ − x₁). Unlike a straight line, a curve has a different gradient at every point, so the tangent must be redrawn each time.
On a straight line, the gradient is constant, so m = (y₂ − y₁) ÷ (x₂ − x₁) works with any two points on the line. On a curve, the steepness keeps changing, so there is no single gradient for the whole curve. Instead, the question asks for the gradient at a specific point, and the only way to answer it from a sketch is to draw a tangent there first.
The most common source of lost marks is not the arithmetic. It is drawing a tangent that is not actually straight, or that crosses the curve instead of just touching it, or reading off two points that are too close together on the tangent, which magnifies small ruler errors into a wrong answer.
Step 1
Mark the given point on the curve
Step 2
Draw a straight tangent touching only there
Step 3
Pick two points on the tangent, far apart
Step 4
Apply (y₂ − y₁) ÷ (x₂ − x₁)
A curve is drawn on graph paper. A tangent has already been drawn touching the curve at the point (2, 5). The tangent also passes through the point (6, 13) on the same line. Find the gradient of the curve at (2, 5).
Solution
(x₁, y₁) = (2, 5) and (x₂, y₂) = (6, 13).m = (13 − 5) ÷ (6 − 2) = 8 ÷ 4 = 2Quick check: a gradient of 2 means the tangent rises 2 units for every 1 unit it runs to the right. Look back at your sketch: does the line climb 2 grid squares for every 1 square it moves across? If the number and the picture disagree, re-check your tangent or your two chosen points.
The curve y = x² is plotted from x = 0 to x = 4. Estimate the gradient of the curve at the point where x = 3.
Solution
m = (15 − 3) ÷ (4 − 2) = 12 ÷ 2 = 6x = 2 and x = 4, that is (2, 4) and (4, 16), has gradient (16 − 4) ÷ (4 − 2) = 6, so 6 is a sensible reading for the steepness in the middle of that stretch. Examiners accept a small range of answers here, because every student's tangent is drawn slightly differently.The step students get wrong
Drawing the tangent so it crosses the curve rather than just touching it at the one point. A tangent should sit on one side of the curve near that point, brushing it, not slicing through it. If your ruler line clearly enters and exits the curve, rotate it slightly until it only grazes the curve at the marked point, then draw the line. Reading two points that are too close together on the tangent is the second common error: small ruler misreads become large errors in the gradient, so always pick two points several grid squares apart.
In word-problem contexts, the gradient of a curve at a point represents how fast one quantity is changing with respect to another at that instant, not on average. This is the same idea used when reading a distance-time graph: on a straight section the gradient gives constant speed, but on a curved section the gradient at each instant must come from a tangent, because the speed itself is changing.
Exam tip: if a question gives you units, for example the curve is "distance in metres against time in seconds", state the gradient with its units too, such as "the speed at t = 3 is 6 m/s". A correct number without the right units and interpretation often loses a mark.
Why is the gradient of a curve different at every point?
A curve bends, so its steepness keeps changing as you move along it. A straight line never bends, so its gradient is the same everywhere. This is exactly why a curve needs a fresh tangent drawn at each point you are asked about, while a straight line only ever needs one gradient value.
Can I use any two points on the curve itself instead of the tangent?
No. Two points taken directly on the curve give the gradient of the chord joining them, which is an average steepness over that stretch of curve, not the gradient at a single point. You must use two points on the tangent line, which touches the curve at only one point.
How long should my tangent line be for an accurate reading?
As long as the graph paper allows. A short tangent forces you to read two points close together, so any small error in where you mark the coordinates has a larger effect on the gradient. Extending the tangent across most of the grid and picking two well-separated points gives a much more reliable answer.
What does a negative gradient on a curve mean?
A negative gradient means the curve is decreasing at that point, the tangent slopes downward from left to right. In a distance-time or similar real-world graph, a negative gradient at a point would mean the quantity is falling at that instant, so read the sign carefully before writing your final answer.
— Mr Gan Math Tuition
Mr. Gan works with students who want a systematic method for reading graphs correctly, not guesswork with a ruler.
Chat with Mr. Gan