Derivatives, Tangent Lines, and Change (Explaining Calculus #5)

We’ve spent a few posts now developing the ideas of limits and continuity, two of the foundational ideas of calculus. We are now going to introduce the third foundational idea, the derivative. The derivative can be thought of as a way to capture the way that things change over time into one single formula.

In order to explain what exactly the derivative is, it will be helpful to first take a detour into a bit of geometry. Since graphs of functions are expressed in the two-dimensional plane, geometry will actually help us out a lot here.

What is a Tangent Line?

The geometric idea we need is called a tangent line. The notion of line should be familiar – a line is something that is straight, and in math we usually assume that lines are both straight and extend forever. Another way you might view a line is the shortest way to travel between two points (this is more of a comment, this view isn’t really necessary in understanding calculus). What then, is a tangent line? Well, a tangent line has to be a tangent line of some other thing. Sometimes, secondary schools will actually define tangent lines for circles, so that’s where I will start. On a circle, a tangent line is a line that “just barely touches” the circle.

What do I mean here? Let’s pause. Visualize a circle – or if you struggle with visualization, draw a circle on some paper. If you draw a random line, then very likely that line will either touch the circle twice or not at all. These lines are not very interesting most of the time. A line that doesn’t touch the circle at all doesn’t even really have a name, and the line that touches the circle twice is called a secant line (I only mention this because secant lines are useful later). The tangent lines are the super-special lines that only touch the circle once. The easiest example to describe would be a horizontal line that just barely touches the top of the circle. That is a tangent line. Take some time to visualize this, or to draw it if you can’t visualize it.

My goal now is to transition this idea of tangent lines for circles to tangent lines for anything at all. Now, for this discussion we don’t care about lines that don’t touch the circle. We thus only care about tangent lines (those that touch the circle exactly once) and secant lines (those that touch the circle exactly twice). For a given line, choose a point P where that line touches the circle. Now, imagine zooming in, and in, and in on that point. Notice now that as we zoom in, if the line is a secant line, we can easily tell the difference between the secant line and the circle, because the don’t really go in the same direction near P. In other words, zooming in really close to a point on the secant line will give an image looking like a crossing of two roads. But tangent lines are not like this. For a tangent line, if you zoom in nearby P, it actually becomes increasingly difficult to differentiate between which one is the circle and which one is the line. For the circle, imagine the horizontal example I mentioned earlier. What I mean here is that if I pick a point on the circle really close to P, then the circle is really, really close to being horizontal at that point.

If this is difficult for you, think more about it. This concept matters a great deal. As it turns out, this distinction actually defines the difference between tangent lines and secant lines. A given line is a tangent line at P if, when we zoom in towards P, the circle and line only ever become harder to tell apart and never easier. A secant line, however, might look a lot like the circle by zooming in just a little bit, but if you zoom in a lot, there will be a clear difference between the two.

The reason this new way of framing things is so helpful is because we don’t have to use circles anymore. We can use any curved shape we like – because the idea of zooming in towards a point has nothing to do with whether or not that shape is a circle. So, going forward, the idea of ‘just barely touching’ the curve is a good definition, but the definition we came up with using ‘zooming in’ is an even better definition for tangent lines.

What is a Derivative?

The idea of the derivative is tightly connected with tangent lines. In fact, I can now define what a derivative is. This is key, so pay attention.

As we’ve discussed before, we can graph functions f(x) on the xy-coordinate plane. As it will turn out, we need f(x) to be continuous in order for any of this to make sense, so assume that f(x) is continuous. What we want to do is to define a totally new function, which we call f^\prime(x) (read this out loud as “f prime of x”). We call this new function the derivative of f(x). Now, suppose that for a given value of x, the line L is the tangent line to the graph of f(x) with coordinates (x, f(x)). Then the definition of f^\prime(x) is that the slope of L is exactly f^\prime(x).

This definition is clear enough – that is, it is unambiguous. But it doesn’t help us very much with actually finding any numerical values of f^\prime(x). We want to know now how we might find actual numerical values for f^\prime(x). This is the task to which we now turn.

This task we are now beginning is the reason I initially defined a secant line. Recall that our goal is to find the slope of the tangent line at the point P on the graph of f(x). For ease, let’s just say the x-coordinate of P is x. Then what we we to do? This is where the continuity of the function f(x) comes into play. Because, if I know that f(x) is continuous, I know that if the x-coordinate of a point Q on the graph of f(x) is really close to the x-coordinate of P, then Q is actually really close to P as well. Phrased differently, for continuous functions, small changes in x-values means small changes in y-values, and since the points P, Q are defined by these two values, which are both small, P and Q must be close by. In fact, let’s say that the x-coordinate of Q is x+h, for some very small (maybe positive, maybe negative) value h.

Now, let’s think about “zooming in” to the point P. Let’s also compare two lines – the tangent line L and the line L^* that travels through both P and Q. Now, if you try some examples yourself (which I highly encourage) then you’ll discover that the closer together P and Q are, then closer together L and L^* are, which means that we have to zoom in further to really tell the difference between L and L^*.

Now, this next step is the reason we spent some much time on limits. What is we let h \to 0 as a limit? Then, in the limit P = Q. This would mean that L and L^* could not longer be isolated, because they would be the same line. Our conclusion now follows:

f^\prime(x) = \text{Slope of } L = \lim\limits_{h \to 0} (\text{Slope of } L^*).

But, the, what is the slope of L^*? Simple – this is the rise-over-run formula. Since the two points we defined are P = (x, f(x)) and Q = (x+h, f(x+h)), then the slope of L^* is \dfrac{\text{rise}}{\text{run}} = \dfrac{f(x+h) - f(x)}{(x+h) - x} = \dfrac{f(x+h) - f(x)}{h}. Therefore,

f^\prime(x) = \lim\limits_{h \to 0} \dfrac{f(x+h) - f(x)}{h}.

We have now defined a legit numerical formula for the derivative in terms of limits. To move away from numbers and back to concepts, what we have done is to say that tangent lines can be approximated by secant lines of super-close-together points. If you are not yet convinced of this, do the following. Draw a very large circle, and draw a point on that circle. Draw a tangent line at that point using a ruler to make sure it is straight. Then, draw a point as close to that other point as you can, then use a ruler to draw a line connecting those two points. You’ll discover that, nearby the circle itself, these lines are very difficult to tell apart.

A Second Way of Writing Derivatives

There are two common ways of writing down derivatives in calculus. The first uses the “prime” notation, which is the f^\prime(x) I’ve been using. But there is also a second way that, although it means exactly the same thing, can in many situations be more convenient. This notation is often used when we have in mind a graph y = f(x) of some function. What, then, is the derivative of y? The notation we’ve used so far is y^\prime, and that is perfectly acceptable. But another way is sometimes useful. Sometimes, we will write down \dfrac{dy}{dx} to talk about the derivative of y. This has the advantage of being very clear about what the x-variable is, and is also convenient when the problem in question is more about graphs than about more abstract functions. There are other situations in which this notation is helpful – those situations will become clear as they arise.

One thing to note. Just because we use the “fraction” notation \dfrac{dy}{dx} does not mean that this is literally a fraction. Although, as we shall see later, the derivative when written in this form does share much in common with ordinary fractions. But it isn’t really a fraction. Care must be taken here. The motivation for this notation is that derivatives are like slopes, and slopes are “changes in y divided by changes in x.” The lowercase d essentially is shorthand for “infinitely small change,” hence the connection of derivatives with limits and this new way of writing.

Also, sometimes I may write \dfrac{d}{dx}[ something ]. This is normally done when writing down some kind of function or graph for something would just be an unnecessary annoyance. This, too, will be used sometimes. For the remainder of the calculus series, the reader should basically think of y', \dfrac{dy}{dx}, and \dfrac{d}{dx}[y] as synonyms and that we can use whichever is most convenient at any moment.

What Do Derivatives Mean?

Change is the essence of what a derivative is. I find that it is most helpful here to rely on examples. If we are talking about position – about where we are – then the derivative of that tells us how are position is changing over time. But that is just what we mean by speed or velocity – changes in location over time. So, the relationship between speed and place embodies the idea of derivatives. If you want to know about speeding up or slowing down – that is, accelerating or decelerating – that is another derivative! Just as you can think of speed as the derivative of position, you can think of acceleration is the derivative of speed.

While numerous other examples can be given, and will be given as I continue in this series, I think this one is the best and clearest. It is also immediately apparent why derivatives are so important. Determining how fast things are going is a very common problem in engineering and physics, and the derivative naturally gives them the tool to study that aspect of our world. There are so many others as well, more than can even be listed, because of how important the notion of things changing over time is in our world. So, if you think change, think derivative.

As we move forward in the series, we will spend some time talking about developing shortcuts for calculating derivatives more quickly. After that, we can enter into the massive world of using derivatives to solve problems.

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