How to Use the Rate of Change Formula in Math and Physics

Rate of change is exactly what it sounds like: how quickly (or slowly) something changes over time. Richard Drury / Getty Images

Do you need to calculate the rate at which something changes over time? Whether it's the change in the x-value over the change in the y-value of a line on a graph, or the distance travelled by a car over the course of an hour-long drive, you'll need a rate of change formula.

In this article, we'll break down rate of change in simple terms.

What Is Rate of Change?

Rate of change (ROC) is exactly what it sounds like: how quickly (or slowly) something changes over time. Usually, you're looking for the average rate of change, or the average rate at which something changes from one point to another. For example:

Distance traveled over time (average speed)
Displacement over time (average velocity)
Velocity over time (average acceleration)
Price over time (financial momentum)

There's also an instantaneous rate of change, or the rate of change at one specific point (rather than between two points). In calculus, the instantaneous rate of change is known as the derivative of a function.

Calculating the Rate of Change

To calculate the rate of change, you divide the change in one quantity by the corresponding amount of change in another quantity. Change is represented as the Greek letter delta (Δ), so the basic formula for rate of change is:

Δ = Δy/Δx

Exactly how you calculate Δy and Δx will depend on the application.

Slope

In algebra, the average rate of change formula is the same as the slope formula, or "rise over run":

Δy/Δx = (y₂ – y₁)/(x₂ – x₁)

Where the rate of change equals the average change of a function between ordered pairs (two points): [x₁, y₁] and [x₂, y₂].

Calculus

In calculus, the rate of change refers to how a function changes between two data points. The formula is:

Δ = (f(b) – f(a))/ b – a

Where the rate of change is equal to the average change in a function between [a, f(a)] and [b, f(b)].

The instantaneous rate of change, or derivative, is equal to the change in a function at one point [f(x), x]:

Δ = f(x)/x

Or

d = dy/dx

Where x is the independent variable, y is the dependent variable and d represents delta (Δ) or change.

Acceleration

The average rate of change of velocity is known as acceleration, and you can calculate it using this formula:

a = (v₁ – v)/t

Where a is acceleration, v₁ is ending velocity, _v0 is starting velocity, and t is time.

Examples

Average Rate of Change of a Linear Function

The average rate of change formula for a function y = f(x) from x = a to x = b is:

A = (f(b) – f(a))/(b – a)

In a linear function (straight line), the rate of change equals the slope of the straight line connecting point (a, f(a)) to point (b, f(b)).

For example, if you were asked to find the average rate of change of the function f(x) = x² − 2x + 4 for the interval [1,3], you would have:

A = (f(3) – f(1))/(3 – 1)

First, let's calculate f(3).

f(3) = 3² – 2(3) + 4

f(3) = 9 – 6 + 4

f(3) = 7

Next, let's find f(a).

f(1) = 1² – 2(1) +4

f(1) = 1 – 2 + 4

f(1) = 3

Now, we can put it all together to find the average rate of change:

A = (7 – 3)/(3 – 1)

A = 4/2

A = 2

You'll notice that the average rate of change is the same as the slope: 2.

Average Acceleration

Calculating the average rate of change for velocity is the same as finding the acceleration.

For example, if you were driving 25 miles per hour (40 kilometers per hour) and increased your speed to 40 miles per hour (64 kilometers per hour) over the course of a minute, the average acceleration would be:

a = (40 miles per hour – 25 miles per hour)/(1/60)

Where v₁ is your final speed (40 miles per hour), v is your starting speed (25 miles per hour) and t is time (1 minute, or 1/60th of an hour).

a = (40 – 25)/(1/60)

a = 15/0.0167

a = 898.2 miles per hour squared (1,437.1 kilometers per hour squared)