Introduction to Derivatives
Rates of change and the slope of a curve
UBC Calculus Online Course Notes

The linear nature of graphs



An example

Let's start with an example. The picture below shows the graph of a function we've seen already: $  y = x^2. $ Notice that at $  x = 1  $ , the graph passes through (1,1). Now let's pretend that we can look at the graph under a microscope and that we can continually increase the magnification of the microscope. Clicking on the "Zoom" button below increases the magnification by a factor of two.

The important thing is that the closer we zoom in, the more the graph looks like a straight line through the point (1,1). We call this line the tangent line to the graph through (1,1).


Questions:
  1. What is the slope of the tangent line?

  2. Since we know that the tangent line passes through the point (1,1), find the equation of the tangent line.

    (Here is a review of straight lines and their equations should you feel you need it.)

  3. Can you think of a function whose graph would not look like a straight line if you zoomed in like this? After you think about this, have a look at this example.

  4. Is the absolute value function differentiable at x = 1?



Important Jargon

We say that a function is differentiable at x if it looks more and more like a straight line near the point x. For instance, the function $  y = x^2  $ is differentiable at $  x = 1,  $ while the absolute value function $  y = |x| 
 $ is not differentiable at $  x = 0. $

If a function $  f(x)  $ is differentiable at a point x, we call the slope of its tangent line the derivative of $ 
f(x)  $ at x. We will write this as

\[  f^\prime(x)  \] .

For instance, the function $  f(x) = x^2  $ is differentiable at x = 1 and the derivative is

\[  f^\prime(1) = 2.  \]


Conclusions

Maybe now you're wondering what the point is behind all of this. That's such a good question that we're going to answer it many times and in many ways throughout this course. You see, straight lines are the graphs of functions which have the form $  y = mx + b  $ and these are exceptionally easy functions to work with. For instance, you only need to know two pieces of information -- a slope and a point through which the graph passes -- to know everything there is to know about such a function. A principal theme in this course is to use information which can be easily extracted from the tangent line to deduce information about the original graph, which could be difficult to deduce more directly.

For instance, in our example above, we know that the slope of the tangent line is 1/2. Since this is positive, we know that the tangent line is rising as we move to the right. What is important is that the graph shares this property with the tangent line; that is, a small increase in the x-coordinate will increase the y-coordinate of the point on the graph. Later in the term, we will see the power of this kind of reasoning.

In this section, we have built a sequence of snapshots of the graph each of which look more like a straight line. However, if we made very careful measurements, we would find that the graphs in the snapshots are still bent just a little bit. However, we can still imagine what the straight line would look like.

In a way, it's like trying to get all the sand out of your car after a day at the beach. At first, there's lots of sand, but after you clean and clean, your car gets closer and closer to being spotless. And while there's always a few specks of sand hiding someplace so that you can never actually have a perfectly clean car, it's all right to think that your car is perfectly clean.



Confused?

If you find this confusing, you shouldn't worry too much (at least for now). These kinds of questions bamboozled some of history's greatest thinkers for several thousand years: ancient mathematicians asked questions like these, but it wasn't until the 1600's that people started to formulate useful answers to them.

What we are really talking about is the fundamental concept of Calculus which we'll soon learn to call a limit . In the next few sections, we'll look at the derivative from a few new angles and find some nifty ways to compute with them. Gradually, the picture will become clear.