Guest Post: Derivatives and an Introduction To Calculus

by Shaun Klassen

One of the mathematics subjects most feared by students is the “dreaded” differential calculus subject.  Absolutely, it is more complicated than more common basic algebra that most would have studied up to this point.  And of course, to work with calculus, one must be familiar with all of the earlier concepts that build up a strong mathematical foundation, including things like algebra, trigonometry, and graphing.  However, this is not to say that calculus has to be hard, or “impossible.”  It is completely doable if you start slowly by learning the general problem solving strategies.  In this guest post, I want to introduce the main concept of differential calculus – the derivative – and I encourage you to visit my math website to find out much more information about this subject.

The name “differential calculus” is a descriptive one – it is based on differences, or changes.  More specifically, it is all about describing how one quantity changes with respect to another one, or in other words, the rate of change.  The derivative is used to express this function, but let’s examine this concept a little more closely by considering everyone’s favourite rate of change: velocity. 

As we all know, velocity is the measure of distance with respect to time.  The simplest way of calculating the velocity is to measure the full distance covered and divide by the length of time it took to traverse that distance.  Essentially, how many meters did you travel in how many seconds, and the results is your velocity in meters per second, or m/s.  (Of course, you can use whatever units you are more familiar with, but this is my post, so we’re talking in meters.  🙂  However, what you need to understand is that this velocity isn’t necessarily the precise one that you would be experiencing at any point along that trip.   What this calculation measures is what we call the “average velocity.”

So, let’s extend this to a real-life, understandable example.  Consider that you are riding a bicycle, and you want to travel from your home to the market that is 1.2 km (1200 m) away.  If it takes you 10 minutes (600 seconds) to get there, your average velocity for the trip is 1200 m / 600 s = 2 m/s.  Now, since you were biking quite quickly, you may think that this is rather slow.  And you are right.  That is because the time that was measured includes all of the time required to accelerate from your home, then you had to stop to wait for traffic to cross the street, and then you talked to a neighbour.  Your average velocity takes into account all of the slow and fast points of your ride.

What about if you consider only portions of your trip?  What if you time yourself to find how long it takes you to travel down your street?  You may come up with numbers such as 200 meters in 50 seconds, which is 200 m / 50 s = 4 m/s.  This seems faster, because you only had to accelerate from your home, and didn’t have very many interruptions before you got to the end of your street.

Now take an even smaller portion, measured while passing the front door of one house until passing the front door of the next house.  This may be something like 20 meters in 4 seconds, which is 5 m/s.  This seems even faster still, and this is because the interval that we are measuring is getting smaller and there is less variation to the velocity within the measured frame.  There was no acceleration or deceleration phases.  It was just constant motion.

Hopefully, you are starting to see the difference between average velocity, and what we call “instantaneous velocity.”

Instantaneous velocity is what you find when you decrease your interval down to an infinitesimally small interval, and your start point and end point approach each other at a single point.  You can say that you are still calculating an average velocity of two points, but the two points are so incredibly close together that they might as well be a single point.

Now let’s bring this a bit closer to what we would see in calculus studies.

Rate of change can obviously be expressed graphically as well.  If you have a graph of distance vs time, the slope of this graph is velocity.  The graph itself can be whatever wavy, irregular function you’d like.  If you take any two points on the graph and draw a line between them, the slope of this line is the average velocity.  And this makes sense, because the curve will waver all around this straight line slope.  However, if you take closer and closer points together, the average velocity will approach the instantaneous velocity.  In fact, you can use limits to express this – as the difference of the two points approaches zero.  What you have when you have a limit like this is a representation of the instantaneous velocity as that point.

This type of limit, where you are assessing rates of change on ever-decreasing intervals, is very common in the fields of math, science, and engineering.  Because of this, this limit is given the name of “derivative,” and you calculate derivatives by the process of “differentiation.”

There is a lot to learn in calculus, and understanding how to find a derivative and knowing how to differentiate are very important parts of it.  This post has been an introduction to the concept, but I have a much more in-depth discussion about derivatives on my site.  Be sure to check that out, and also visit my home page at Math Concepts Explained to search for other math topics of interest.

My thanks to Guillermo for letting me guest post here on his fantastic blog!