# Find the instantaneous rate of change

Finds the instantaneous rate of change f′(a) of a function at x = a, exactly and as a decimal, checked numerically.

- Page: https://www.acalculator.org/math/instantaneous-rate-of-change-calculator
- JSON spec: https://www.acalculator.org/math/instantaneous-rate-of-change-calculator.json
- Version: f632eac5207e

## Default answer

Example with the default inputs (Function f(x) 3x^2 - 4x + 1, At x = a 2): The instantaneous rate of change of 3x^2 - 4x + 1 at x = 2 is 8.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| f | Function f(x) | The quantity as a function of x, typed like 3x^2 - 4x + 1 or sin(x). |
| a | At x = a | Where the rate is taken: a number or a constant such as pi/2. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| rate | Rate of change f′(a) | The derivative of f at a, written exactly. |
| value | As a decimal | f′(a) to 10 significant figures. |
| derivative | Derivative f′(x) | The derivative of f at every x. |

## Method

The rate of change at a is the derivative f′(a) = lim (f(a + h) − f(a))/h as h → 0. A computer algebra system finds it; it is shown only when it matches difference quotients.

## Assumptions

- The variable is x; angles are in radians; ln is the natural logarithm.
- An answer that fails its check is not shown.

## Worked examples

1. f = 3x^2 - 4x + 1, a = 2 gives rate = 8, value = 8, derivative = 6x - 4. Source: OpenStax, Calculus Volume 1, section 3.1 Defining the Derivative, Example 3.5. https://openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative.
2. f = 0.4x^2 - 4x + 70, a = 3 gives rate = -8/5, value = -1.6.

## FAQ

### What is the instantaneous rate of change?

It is how fast f is changing at one exact value of x: the derivative f′(a) = lim (f(a + h) − f(a))/h as h → 0. On the graph it is the slope of the tangent line at (a, f(a)).

### What is the difference between average and instantaneous rate of change?

The average rate of change from a to b is (f(b) − f(a))/(b − a): the slope of the secant line. The instantaneous rate is the limit of that average as b gets closer to a. For f(x) = x² from 3 to 3.1 the average is 6.1; the instantaneous rate at 3 is 6.

### Is instantaneous velocity the same thing?

Yes. If s(t) is a position at time t, the instantaneous velocity at t = a is s′(a), the instantaneous rate of change of position. Type the position with x in place of t: for s = 16t², type 16x^2.

### What are the units of the answer?

Units of f per unit of x. If f is a temperature in °F and x is time in hours, f′(a) is in °F per hour; a negative value means f is falling at a.

### Why is there no rate of change at some points?

The rate is the slope of the tangent line, and some functions have none at some points. |x| has a corner at 0 (slope −1 on the left, 1 on the right), and the cube root of x has a vertical tangent at 0. The page compares slopes just left and right of a and gives no rate when they differ.

### How is the answer checked?

A computer algebra system finds the derivative. It is compared with a numeric difference quotient at 20 points, and the rate at a with the slope of f just left and right of a. If a check fails, the page says "No verified answer".

## Sources

- OpenStax, Calculus Volume 1, section 3.1 Defining the Derivative: https://openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative