Before You Watch

This video continues directly on from Rates of Change and Differentiation, in which the concept of a rate of change was introduced and we investigated the need to know the value of instantaneous rates of change. The term ‘to differentiate’ was also discussed. Differentiation is the process of calculating a rate of change. This video describes how to differentiate a specific class of equations known as polynomials. So make sure you’ve seen Rates of Change and Differentiation recently before watching this video.

This video also builds upon earlier algebraic concepts such as indices, including negative indices, and linear equations. It is important to be comfortable with algebra and manipulating algebraic equations before continuing with calculus, so watch those videos again if you need to, then come back.

The Video

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Some Practice Questions


Now What?

By now you will be familiar with the three videos that introduce the differential branch of calculus: Introduction to Calculus, Rates of Change and Differentiation and this topic, Differentiation of Polynomials. You should understand the core concepts of calculus and know what a rate of change is. You will also know that differentiation is all about calculating the rate of change, and know how to differentiate one category of equations, the polynomials. From here there are two main directions you can go.

One option is to explore how to differentiate other types of equations, such as those involving trigonometry, or exponentials. To do this you should consider looking at sites such as the Khan Academy:


Alternatively, you could investigate the other branch of calculus, integral calculus. This is introduced in the video Integration.

But When I am Going to Use This

Calculus is the mathematical study of how things change relative to one another. For instance, velocity (or speed) is a change of position over a change in time, and acceleration is a change in velocity over a change in time – so any motion is studied using calculus. Other examples include the flow of water through pipes over time, or changing commodity prices against demand. Because change is everywhere, the potential applications for calculus are endless, particularly in engineering and science. Calculus is necessary knowledge for any degree related to engineering or science.

Other Links

Maths Is Fun has a great page that takes you through a simple problem which highlights the need for calculus to discuss changes happening around us. It then continues to explore the main two areas of calculus, differentiation and integration, and provides regular questions to test your understanding. 

IntMath gives some historical perspective to explain the sometimes confusing notation that is used in calculus, discussing how it is the mixed product of two mathematicians working independently. It also provides some excellent examples of applications of calculus that are in common use today, as well as helpful applets to understand both differential and integral calculus.

The Khan Academy has a comprehensive set of video tutorials covering a wide range of mathematical and other concepts, as well as questions to test your knowledge. This content provides a whole chapter on taking the derivatives, including of harder equations not covered in this video.

Patrick JMT (Just Maths Tutorials) has an extensive set of video tutorials covering a large range of mathematical concepts. This content runs through differentiation of simple polynomials, but the site also provides videos demonstrating more complex differentiation.