Rate of Change Calculator

The average rate of change is the change in the output divided by the change in the input: (y₂ − y₁) ÷ (x₂ − x₁). For the points (2, 5) and (6, 17) the output rises by 12 while the input rises by 4, so the rate of change is 12 ÷ 4 = 3. This calculator handles three versions of that question: two plain points, a polynomial function over an interval, and a percent rate of change per year, month, week or day.

Three neighboring tools answer closely related questions. The Slope Calculator uses the same arithmetic but frames the answer as the geometry of a line, with the y-intercept, the equation and the angle of incline. The Percentage Change Calculator gives the total percent difference between two numbers when there is no time axis at all. The CAGR Calculator is the finance-specific form of the compound rate below. Use this page when you want the rate itself, with the units and the interval attached.

The rate of change formula

Rate of change answers a single question: how much does the output move for each unit the input moves? The formula is the same whether you call the inputs x-values, hours, weeks or years:

Written for a function over the interval from a to b, exactly the same idea reads:

The numerator is often written as Δy (delta y, the change in the output) and the denominator as Δx (delta x, the change in the input), which is why the rate of change is also read aloud as "delta y over delta x" or "rise over run". Nothing about the formula assumes a straight line: it works on curves, on scattered readings and on a table of measurements, because it only ever looks at the two endpoints.

Worked example: two points

Take the points (2, 5) and (6, 17), the values loaded in the calculator above.

1. Change in the output: 17 − 5 = 12.
2. Change in the input: 6 − 2 = 4.
3. Average rate of change: 12 ÷ 4 = 3.
4. Line through both points: b = 5 − 3 × 2 = −1, so y = 3x − 1.

The plain reading is that y goes up by 3 units for every 1 unit that x goes up. Expressed as a percentage instead, the output grew from 5 to 17, a total increase of 240% across the interval, which is 60% per unit of x. The absolute rate and the percent rate answer different questions, so the calculator reports both.

Worked example: a function over an interval

Now take the quadratic f(x) = x² + 3x − 5 on the interval from a = 1 to b = 4.

1. f(1) = 1 + 3 − 5 = −1.
2. f(4) = 16 + 12 − 5 = 23.
3. Change in the output: 23 − (−1) = 24.
4. Width of the interval: 4 − 1 = 3.
5. Average rate of change: 24 ÷ 3 = 8.

That 8 is the slope of the straight secant line joining the two points on the parabola, namely y = 8x − 9. The curve itself is nowhere near constant: its derivative is f′(x) = 2x + 3, which equals 5 at x = 1 and 11 at x = 4. The single number 8 is the honest average of that whole range, and for a quadratic it is reached exactly at the midpoint, x = 2.5, where f′(2.5) = 8.

Step rates across the same interval

Splitting the interval from 1 to 4 into six half-unit steps shows how much the curve speeds up on the way, even though the six step rates still average out to 8:

The six step rates run from 5.5 up to 10.5, and their average is 48 ÷ 6 = 8, matching the overall figure. This is the single most useful habit when reading any rate: ask how wide the interval was before you trust the number.

Average versus instantaneous rate of change

Shrinking the interval toward one point turns the average rate into the instantaneous rate, which is what a derivative measures. Starting at x = 2 on the same quadratic, whose derivative there is f′(2) = 7:

The averages march steadily toward 7 without ever quite arriving, which is the whole idea behind a limit. In a calculus course the average rate of change is usually taught first precisely so this table makes sense later: the derivative is the value the secant slopes approach as the second point slides into the first.

Percent rate of change

Sometimes the absolute rate is less useful than the relative one. A gain of 100 units means something different starting from 200 than from 20,000. The percent rate of change divides the change by the starting value:

Spreading that total evenly gives the simple percent rate per period, while the compound rate is the constant percentage that, applied on top of itself each period, actually reproduces the ending value:

The two answers differ, and the gap widens as the total change grows. Starting from 1,000 across 5 periods:

Doubling from 1,000 to 2,000 in five periods is 20% per period on a simple reading but only 14.87% compound, because each compound step is applied to a bigger base than the one before it. Quote the simple rate for a plain summary and the compound rate whenever the quantity grows on top of itself.

How to use this calculator

1. Pick a mode. Two points for raw coordinates or two readings, Function f(x) for a polynomial over an interval, Percent change for a value tracked over time.
2. Two points: type x₁, y₁, x₂ and y₂. The x-values are the inputs (time, distance, quantity) and the y-values are the outputs you measured.
3. Function mode: type the polynomial using the caret for powers, such as x^2 + 3x - 5, then set the interval endpoints a and b.
4. Percent mode: enter the starting value, the ending value and how many periods passed, then choose years, months, weeks or days.
5. Read the headline number at the top of the result card, then check the step-by-step panel below it to see exactly which subtraction and division produced it.
6. Scroll the table to see the interval broken into steps, or the value at each period under the compound rate.

Everything recalculates as you type, so changing one endpoint and watching the rate move is often more instructive than the single answer.

Who this calculator is for

• Algebra and pre-calculus students checking homework on average rate of change over an interval.
• Calculus students comparing the secant slope with the derivative and watching one approach the other.
• Lab and science students turning a table of timed readings into a rate with the correct units attached.
• Analysts and operators converting a before-and-after pair into a monthly or yearly rate they can quote.
• Anyone tracking progress, from pounds per week to dollars per year, who wants the pace rather than the total.

Key terms explained

• Average rate of change: the change in output divided by the change in input over an interval. Needs exactly two points.
• Instantaneous rate of change: the rate at a single input, found as the derivative. It is the limit of the average rate as the interval shrinks to nothing.
• Secant line: the straight line drawn through two points on a curve. Its slope is the average rate of change between them.
• Tangent line: the line touching a curve at one point. Its slope is the instantaneous rate of change there.
• Delta (Δ): shorthand for "change in". Δy is the change in the output, Δx the change in the input.
• Interval: the stretch of inputs from a to b that the average covers. Written as [a, b] in textbooks.
• Percent rate of change: the change expressed as a share of the starting value, then spread across the periods.

Rates of change in real units

The formula never changes, only the labels on the two axes. Each of these comes from the same division:

The last row is a good illustration of why both rates matter: $120 per year is the absolute pace, the total increase is 30%, and the compound rate is 9.14% per year. All three describe the same pair of numbers.

What changes the result the most

• The width of the interval. A wide interval smooths out everything that happened inside it. The step table above shows rates from 5.5 to 10.5 hiding behind a single average of 8.
• Which two endpoints you pick. On a curve, moving one endpoint changes the answer even though the function did not change at all.
• The starting value, for percent rates. The same absolute gain is a huge percentage off a small base and a rounding error off a large one.
• Simple versus compound. For a doubling over five periods the two readings differ by more than five percentage points.
• The units on each axis. Reading a rate per month when the inputs were weeks is off by a factor of about 4.3.

Tips for getting it right

• Write the units into the answer immediately. "8" means little; "8 dollars per year" cannot be misread.
• Keep the same point first in the numerator and the denominator. Flipping only one of them flips the sign.
• Subtract carefully with negatives: 23 − (−1) is 24, not 22.
• On a curve, quote the interval with the rate. "8 on [1, 4]" is a complete answer; a bare "8" is not.
• When the data grows on top of itself, report the compound rate and say so, rather than dividing the total by the number of periods.
• Sanity-check the sign before anything else. A falling quantity must produce a negative rate.

Limitations and assumptions

• The average rate of change only looks at the two endpoints. Everything in between, including peaks, dips and reversals, is invisible to it.
• The function mode accepts polynomials with whole-number exponents from 0 to 10. Trigonometric, exponential, logarithmic and rational functions are outside its range.
• A rate is not a forecast. Extending it past the interval assumes the pattern continues, which the arithmetic itself never promises.
• Percent rates need a non-zero starting value, and the compound rate needs both values to be positive, since a negative base has no real root.
• Results are rounded for display. Internally the calculation carries full floating-point precision.

How it compares to related calculators

This page answers "how fast is the output changing per unit of input?" If your question is slightly different, a sister tool fits better:

• For the geometry of a line, its equation, y-intercept and angle, use the Slope Calculator.
• For the total percent difference between two numbers with no time axis, use the Percentage Change Calculator or the Percentage Increase Calculator.
• For the compound annual growth rate of an investment or a balance, use the CAGR Calculator.
• For distance over time specifically, with unit conversion built in, use the Velocity Calculator.
• To turn a rate and one point into the full equation of the line, use the Point-Slope Form Calculator.
• To summarize a whole list of readings rather than two endpoints, use the Average Calculator.

Sources

Every figure on this page is deterministic mathematics rather than sourced data. The average rate of change, (f(b) − f(a)) ÷ (b − a), is an exact definition from algebra and calculus, as is the derivative used to compare the average with the instantaneous rate; both are computed here with standard arithmetic and need no external source. The percent and compound-rate formulas are likewise exact definitions. The worked examples, the step table, the shrinking-interval table and the percent table on this page were each computed directly from those formulas rather than quoted from anywhere. The one place external standards apply is the units in the real-world table: the mile, the pound and the degree Fahrenheit used there follow the U.S. customary definitions maintained by NIST Special Publication 811, Guide for the Use of the International System of Units. No market rates, tax figures or health guidelines are involved.