instantaneous rate of change example and guide

Find Instantaneous Rate of Change Example and Guide

Finding the instantaneous rate of change is one of the most fundamental skills you will learn in your studies of differential calculus. Developing a good understanding of how to find the instantaneous rate of change of a function will help set you up for success throughout your study of calculus!

I have put together this instantaneous rate of change example guide to break this difficult concept up into easier to understand pieces. My hope is that this explanation makes your study of calculus easier!

What is Instantaneous Rate of Change?

In simple terms, the instantaneous rate of change helps us understand how a given function is changing at a given instant. This is done by measuring the slope of a tangent line of a curve at a specific point. 

A tangent line is a straight line that touches the curve of a given function at a single particular point. 

The concept of instantaneous rate of change is used to analyze the behavior of functions, solve optimization problems, and understand the dynamics of changing quantities at an exact moment in time.

What is a Derivative?

In your studies of calculus you will spend a lot of time analyzing how to find the derivative of a function. You will use strategies such as the power rule, product rule, and chain rule to find expressions for the derivative of a given function.

But what exactly is a derivative? And most importantly, what is the connection between derivatives and instantaneous rates of change?

As it turns out, derivatives have a very important connection to instantaneous rates of change! 

Derivatives provide us with an expression for the instantaneous rate of change at a given point. This means that a derivative is essentially a formula for the slope of the tangent line (also known as the gradient of the tangent).

When we find the value of the derivative at a specific point we are able to determine the slope of the line tangent to the function at that given point. This is convenient as it allows us to find the instantaneous rate of change at different values on the graph of a given function.

What’s the Difference Between Instantaneous Rate of Change and Average Rate of Change?

Average rates of change help us calculate the slope of a line joining two points. By definition, average rate of change is the slope of the secant line passing through two points. Because of this, we say that average rates of change are calculated across an interval.

By comparison, instantaneous rates of change are calculated over very, very small intervals. In mathematical terms, instantaneous rate of change is the limit of the average rate of change as the interval over which the change is measured approaches zero.

But that’s all really scary and complicated sounding, isn’t it? Let’s take a closer look at this idea.

What is the General Formula for the Instantaneous Rate of Change?

The instantaneous rate of change formula looks really complicated, but at its core it is really similar to the formula for the slope of a line. 

Yes, the slope of a line formula you learned before you even knew what calculus was!

Remember this formula?

$$slope = \frac{delta \, y}{delta \, x}$$

The slope of a line can be calculated using delta y over delta x. This is essentially the formula for average rate of change of a function. It is the slope of a line over an interval made up of two points.

This formula helps us calculate the slopes of secant lines (lines that join two points on the graph of the function).

But if we are calculating instantaneous rate of change, we are not interested in the slope of the secant line passing through two points! Instead, we want to know the slope of a line at a particular point at a specific instant!

a line drawn through two points on a curve and a line drawn through one point on a curve

That’s where this big scary instantaneous rate of change formula comes in!

$$f'(x)=\lim_{h \to 0}  \frac{f(x+h)-f(x)}{h}$$

But wait, don’t you need two points to find the slope of a line? 

Technically, yes. And the instantaneous rate of change formula still does that. This formula allows us to determine a really good approximation for the slope at a single point. 

While it looks complicated, this formula is really just the same as the formula for the slope of a line using a smaller interval. You are just using two points that are really close together. Infinitely close together, in fact!

The good news is we rarely have to use this complex formula to find the instantaneous rate of change of a function. The instantaneous rate of change example below will help us find out more!

Instantaneous Rate of Change Example

Now that you understand the concept of instantaneous rate of change, let’s take a look at an instantaneous rate of change example. Let’s find the instantaneous rate of change of the function f shown below.

Consider the following function:

$$f(x)=2x^3-x^2+1$$

Since the function is a polynomial function, we can apply the power rule for derivatives to determine an expression for the instantaneous rate of change at a particular instant.

Recall that the power rule tells us to bring the exponent on each term in front of the variable and reduce the power by 1. Doing this results in:

$$f'(x)=6x^2-2x$$

When we found the derivative of the first term, we multiplied the exponent of 3 by the coefficient of 2. Notice that the derivative has a degree of one less than the original function.  

Remember that the derivative of 1 is just zero. This is because the derivative of any constant is always zero since constants have no rate of change!

Next, let’s use this derivative function to find the instantaneous rate of change of the function f at x=1. To do this, we substitute x=1 into the derivative function: 

$$\begin{split} f'(1)&=6(1)^2-2(1) \\\\  &=6-2 \\\\\ &=4  \end{split} $$

Therefore we can say that the instantaneous rate of change of the function f at x=1 is 4. Remember that this is the slope of the curve at the specific instant when x is equal to 1.

Understanding Instantaneous Rate of Change

Instantaneous rate of change is a complex sounding topic that is crucial to understand in the study of calculus. In fact, this formula is one of the reasons many students find calculus so challenging

However, if you look closely at this topic, it is really nothing more than finding the slope of a line with smaller time intervals!

The important part is to recognize the connection between the derivative of a function and its instantaneous rate of change. 

While it is important to understand the concept of instantaneous rate of change, we rarely apply the complex first principles definition of the derivative. Instead, we use strategies such as the power rule, product rule, and chain rule to help us efficiently find expressions for a function’s derivative.

My hope is that the above example and the explanations I have provided here help you understand and apply the concept of instantaneous rate of change.

There are so many practical applications of rates of change in the real-world. Whether you are calculating instantaneous velocity or determining the marginal profit for the sale of a product, the concept of instantaneous rate of change is essential.

Hopefully the previous example provided you with an introduction to how this important concept can be used in differential calculus!

If you are looking to continue your studies of differential calculus, check out my walkthrough of the first derivative test!

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