Differential Calculus: Definition & Applications

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  • 0:03 Average and…
  • 1:32 Definition
  • 1:55 Rates of Change
  • 4:59 Derivatives & Differentiation
  • 5:50 Lesson Summary
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Lesson Transcript
Instructor: Shaun Ault

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

This lesson is an introduction to differential calculus, the branch of mathematics that is concerned with rates of change. If you ever wanted to know how things change over time, then this is the place to start!

Average & Instantaneous Velocity

What is differential calculus? Let's take a car trip and find out!

Suppose we take a trip from New York, NY to Boston, MA. That's roughly 200 miles, and (depending on the traffic), it will take about four hours. Now, we all know that distance equals rate multiplied by time, or d = rt. In this example, we have distance and time, and we interpret velocity (or speed) as a rate of change. So we could figure out our average velocity during the trip by dividing distance over time. That's:

(200 miles) / (4 hours) = 50 miles/hour

However, if you've ever driven in Boston or New York, you know that when you're in the city, you're driving much slower than 50 mph! On the other hand, when you get to the highways, you expect to be going more than 50 mph. The calculation we did is just an average, and it's answering the question: if my velocity stayed the same throughout the entire trip, what would it be?

But your car knows better. It has a speedometer that keeps track of the speed (velocity) at any given instant. When you look at the speedometer and it reads 61 mph, that is telling you the instantaneous velocity at the particular instant of time you decided to look at it. How does it know? Your car is doing something like differential calculus to figure it out!


Differential calculus is the study of rates of change of functions, using the tools of limits and derivatives.

Now I know some of these words may be unfamiliar at this point in your journey, but we'll take some time to explain them in this lesson. Remember, this is only an introduction to the important topic of differential calculus. A full treatment takes a semester or more to explain!

Rates of Change

Okay, let's dig into these ideas further. There are two related concepts to discuss, average rate of change and instantaneous rate of change.

Average Rates

Average rates start with the idea of a y = f(x). You input x, and get back output y. Let's assume you know this from algebra or pre-calculus. In calculus, a rate of change is a measurement of how the y-values of a function change with respect to changes in the x-values. Sound confusing? Well, let's just think of rate in the same way as velocity. It's a fraction, with the change in y on top divided by the change in x on the bottom.

Suppose we start at x1 and end at x2. Then the change in x is the difference, x2 - x1, which is similar with the y-values as we see here. The triangle notation is a Greek capital delta, which means 'change in' a quantity.

Change in y over change in x

This is really nothing new. It's the same slope formula you've seen way back in algebra class! It's just rise over run. And it measures what we call the average rate of change of f on the interval from x1 to x2. The only difference now is that we have to use the given function f to find the y-values, like this:

Average Rate of Change Formula

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