The Gradient of a Curve

One gradient for a straight line

The straight line y = 3x + 1 rises 3 for every 1 it goes across. From (0, 1) to (1, 4) the rise is 3 and the run is 1. From (1, 4) to (3, 10) the rise is 6 and the run is 2, and 6 ÷ 2 = 3 again.

Any two points on a straight line give the same gradient, so one number, 3, describes the steepness of the whole line.

The line y = 3x + 1 through (0, 1), (1, 4) and (3, 10). A run of 1 gives a rise of 3, and a run of 2 gives a rise of 6: the gradient is 3 everywhere.

A curve changes its steepness

The curve $y=x^2$y = x² is different. Step across 1 at a time and join the points with chords. From (0, 0) to (1, 1) the curve rises 1. From (1, 1) to (2, 4) it rises 3. From (2, 4) to (3, 9) it rises 5, and from (3, 9) to (4, 16) it rises 7.

Each run is 1, so the gradients of those chords are 1, 3, 5 and 7. The curve is nearly flat near the bottom and gets steeper further out. No single number describes its steepness, so "what is the gradient of $y=x^2$y = x²?" has no answer as it stands.

The curve $y=x^2$y = x² through (0, 0), (1, 1), (2, 4) and (3, 9), with a step of 1 across under each piece. The rises are 1, 3 and 5, so the curve gets steeper as x grows.

The gradient at one point

So ask a smaller question: how steep is the curve at one point? Take the point (2, 4). The line that just touches the curve there, without cutting across it, is the tangent at (2, 4). The gradient of the curve at a point is defined to be the gradient of the tangent at that point.

At (2, 4) the tangent is the line y = 4x − 4. It passes through (1, 0) and (3, 8), so it rises 8 over a run of 2, and its gradient is 8 ÷ 2 = 4. The gradient of $y=x^2$y = x² at (2, 4) is 4.

To check that this line touches the curve and does not cross it, subtract: $x^2-\left(4x-4\right)=x^2-4x+4={\left(x-2\right)}^2$x² − (4x − 4) = x² − 4x + 4 = (x − 2)². A square is never negative, so the curve is never below the line. It equals 0 only at x = 2, so the curve meets the line at (2, 4) and nowhere else.

The tangent to $y=x^2$y = x² at (2, 4) is y = 4x − 4. From (1, 0) to (3, 8) it rises 8 over a run of 2, so the gradient at (2, 4) is 4.

A different gradient at every point

The same check works at other points. At (1, 1) the tangent is y = 2x − 1, because $x^2-\left(2x-1\right)={\left(x-1\right)}^2$x² − (2x − 1) = (x − 1)², and its gradient is 2. At (3, 9) the tangent is y = 6x − 9, because $x^2-\left(6x-9\right)={\left(x-3\right)}^2$x² − (6x − 9) = (x − 3)², and its gradient is 6.

So the gradient of $y=x^2$y = x² is 2 at x = 1, 4 at x = 2 and 6 at x = 3. At the bottom, (0, 0), the tangent is the x-axis, y = 0, with gradient 0. To the left of the bottom the curve falls: at (−1, 1) the tangent is y = −2x − 1, with gradient −2.

The gradient is a different number at each point, so it is a function of x. Each value here is twice x. Two lessons on, first principles shows why, without being told the tangent first.

Two tangents to $y=x^2$y = x². The tangent at (1, 1) is y = 2x − 1, with gradient 2. The steeper tangent at (3, 9) is y = 6x − 9, with gradient 6.

A rate at one instant

When a graph shows how something changes over time, its gradient is a rate of change. The gradient of a chord is an average rate over an interval. The gradient of the tangent is the rate at one instant.

A ball dropped from a tower falls $y=5x^2$y = 5x² meters in x seconds. Its distance-time graph is a curve like $y=x^2$y = x², stretched upward. The gradient of a chord is the ball's average speed over that time, and the gradient of the tangent at a time is its speed at that moment, the reading a speedometer would show.

The usual mistakes

Saying that a curve has one gradient. Only a straight line has the same gradient everywhere. On $y=x^2$y = x² the gradient is 2 at x = 1 and 6 at x = 3.

Thinking two points are equally steep because they are close together on the page. On $y=x^2$y = x² the curve is steeper at x = 3 than at x = 1, because the tangent at x = 3 rises 6 for each 1 across and the tangent at x = 1 rises only 2.

Giving the height for the gradient. At (3, 9) the height of the curve is 9, but its gradient there is 6, the gradient of the tangent.

Taking a chord across a wide interval as the gradient at a point. The chord from (2, 4) to (3, 9) has gradient 5, but the gradient at (2, 4) is 4.

A dropped ball

In the application below, the ball falls $y=5x^2$y = 5x² meters in x seconds. The average speed over each interval is the gradient of a chord, and a steeper chord later on shows the ball falling faster and faster.