# Instantaneous Rate Of Change Formula

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# Instantaneous Rate Of Change Formula

**Instantaneous Rate Of Change: **The rate of change at one known instant is the Instantaneous rate of change, and it is equivalent to the value of the derivative at that specific point. So it can be said that, in a function, the slope, **m **of the tangent is equivalent to the instantaneous rate of change at a specific point. One more method to comprehend this concept clearly is with the difference quotient and limits. The average rate of change of **y **with respect to **x **is the difference quotient. Now if one looks at the difference quotient and lets Delta **x->0**, this will be the instantaneous rate of change. In guileless words, the time interval gets lesser and lesser.

Imagine that you drive to a grocery store 10 miles away from your house, and it takes you 30 minutes to get there. That means that you traveled 10 miles in 1/2 hour, at an average speed of 20 miles per hour. (10 miles divided by 1/2 hour = 20 miles per hour). The speed of your car is a great example of a **rate of change**.

## Instantaneous Rate Of Change Calculator

A rate of change tells you how quickly something is changing, such as the location of your car as you drive. You can also measure how quickly your hair grows, how much money your business makes each month, or how much water flows over a dam. All of these, and many more, can be represented by calculating the average rate of change of a quantity over a certain amount of time.

One easy way to calculate a rate of change is to make a graph of the quantity that is changing versus time. Then you can calculate the rate of change by finding the slope of the graph, like this one. The slope is found by dividing how much the *y* values change by how much the *x* values change. Let’s look at a graph of position versus time and use that to determine the rate of change of position, more commonly known as speed.

Let’s go back a moment and think about that grocery store trip again. We calculated that your average speed for the entire trip was 20 miles per hour, but does that mean that you were traveling at exactly 20 miles per hour for the entire trip? What about when you were stopped at a red light or were stuck in traffic that wasn’t moving? During those times you weren’t moving at all, so your speed was zero.Instantaneous vs Average Rate of Change

When you measure a rate of change at a specific instant in time, this is called an **instantaneous rate of change**. An **average rate of change** tells you the average rate at which something was changing over a longer time period. While you were on your way to the grocery store, your speed was constantly changing. Sometimes you were moving faster than 20 miles per hour and sometimes slower. At each instant in time, your instantaneous rate of change would correspond to your speed at that exact moment.

### Calculating from a Graph

So, we saw that you could calculate the average rate of change by calculating the slope of a line, but does that work for instantaneous rates of change as well? In fact, it does, although you have to think about slope a little differently than you may have before.

If you have a graph of your position vs. time that is NOT a straight line and you want to calculate your instantaneous speed, you can draw a line, known as a **tangent line**, that only touches the graph at one point. The slope of this tangent line will give you the instantaneous rate of change at exactly that point.

As you can see from the calculation on this graph, *v* equals 20 meters divided by 5 seconds minus 1.5 seconds, meaning 3.5 seconds, which equals 5.7 meters per second. How does that compare to the average rate of change? To determine your average speed over the whole trip, calculate the slope of a line drawn from the first point on the graph to the last point.

As you can see from the calculations on this graph, our average speed equals 49 meters divided by 7 seconds, which equals 7 meters per second. That is our average speed.

### Calculating with Calculus

When we found your instantaneous speed using a graph, we drew a tangent line that only touched the graph at one point, and then calculated the slope of that line to find the rate of change. In calculus, the slope of a line tangent to a graph is called a **derivative**. So, if you have an equation that describes the position of the car, you can find the derivative, and this will give you a new equation for the speed of the car at every instant in time.

## Instantaneous Rate Of Change Formula

Recall that the average rate of change of a function y = f(x)

on an interval from x1 to x2 is just the ratio of the change in y to

the change in x:

∆y∆x= f(x2) − f(x1) x2 − x1.

For example, if f measures distance traveled with respect to time

x, then this average rate of change is the average velocity over that

interval. But that leaves us with the question of what is the instantaneous velocity at some moment x0, the velocity that the

speedometer in a car is claimed to give us?

The answer is in some sense quite easy to give: The instantaneous rate of change of the function y = f(x) at the point x0

in its domain is:

lim

x→x0

∆y∆x

= lim

x→x0

f(x0) − f(x) x0 − x.

provided this limit exists.

Example 1. Let f(x) = 1/x and let’s find the instantaneous

rate of change of f at x0 = 2. The first step is to compute the

average rate of change over some interval x0 = 2 to x; and in order

for this to make sense we need x 6= 2. So that average rate of change

is

∆y ∆x = f(2) − f(x) 2 − x

2 − x =x − 2 2x(2 − x)

= −12x.

Thus, the instantaneous rate of change at x0 = 2 is

lim

x→2 ∆y∆x= lim

x→2−12x=14.

The instantaneous rate of change at some point x0 = a involves

first the average rate of change from a to some other value x. So if

we set h = a − x, then h 6= 0 and the average rate of change from

x = a + h to x = a is

∆y∆x=f(x) − f(a)x − a

= f(a + h) − f(a)h.

Either of these last two ratios is known as a difference quotient, a term we shall us repeatedly. With this notation the instantaneous rate of change of f at x = a is the limit, if it

exists,

lim

h→0

f(a + h) − f(a)h.

## How To Find Instantaneous Rate Of Change

The **derivative** tells us the rate of change of one quantity compared to another at a particular instant or point (so we call it “instantaneous rate of change”). This concept has many applications in electricity, dynamics, economics, fluid flow, population modelling, queuing theory and so on.

Wherever a quantity is always changing in value, we can use **calculus** (differentiation and integration) to model its behaviour. In this section, we will be talking about events at certain times, so we will be using* *Δ*t* instead of the Δ*x* that we saw in the last section Derivative from First Principles.

**Note: **This section is part of the introduction to differentiation. We learn some (much easier) rules for differentiating in the next section, Derivatives of Polynomials.

We learned before that velocity is **distance divided by time**. But this only works if the velocity is **constant**. We need a new method if the velocity is changing all the time.

Using the power rule for derivatives, we end up with 4x as the derivative. Plugging in our point’s x-value, we have:

4(1)=4

This tells us that the slope of our original function at (1,6) is 4, which also represents the instantaneous rate of change at that point.

If we also wanted to find the equation of the line that is tangent to the curve at the point, which is necessary for certain applications of derivatives, we can use the Point-Slope Form:

y−y1=m(x−x1)

with m = slope of the line.

Plugging in our x,y, and slope value, we have:

y−6=4(x−1)

## How do you calculate instantaneous rate of change?

**estimation**of the

**instantaneous rate of change**of a function at a point, draw a line between two points (“reference points”) very close to your desired point, and

**determine**the slope of that line. You can improve the accuracy of your

**estimate**by choosing reference points closer to your desired point.

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

**instantaneous change**is defined as the derivative of the function evaluated at a single point. The

**average change**is the same as always (pick two points, and divide the

**difference in the**function values by the

**difference in the**x values).

## Is the instantaneous rate of change the derivative?

**Derivative**as an

**Instantaneous Rate of Change**. The

**derivative**tells us the

**rate of change**of one quantity compared to another at a particular instant or point (so we call it “

**instantaneous rate of change**“).