Maxima and Minima
UBC Calculus Online Course Notes

What does a derivative tell us about a function?

You are probably wondering what information the derivative of a function gives us about a function we might be interested in. That's what we will discuss on this page.

We have already learned that the derivative of a function tells us a lot about what happens when we inspect the graph of a function with a powerful microscope: specifically, it tells us how steep the tangent line to the graph would be at the point we are zooming in on.

We also know that that steepness, or slope varies from point to point: some parts of the graph are flatter, and some parts slope up more steeply.

On the graph below, we have plotted a graph of a familiar function,

y=f(x) = x3-x

(You might remember that this is a polynomial, and that we have figured out how the bumps appear on this graph when the term with the x appears with a minus sign. If you forgot this, have a look at polynomials.)

We have decorated the graph of this polynomial with a few of its tangent lines. You can see by looking at these lines that their slope changes from point to point.

Now lets take a look at the slopes of the tangent lines we encountered on the graph. Here is what they might look like if we just string them up along the x axis and forget the exact point at which they were attached to the graph. Let's pretend for a minute that we have forgotten what the original function looked like exactly; but here is a record of all the "ups and downs" that it had.

In fact, we can even simplify our record further and forget about the exact values of the slopes of these tangent lines, and just record their signs. (We used the abbreviation "neg" below, because "-" looked too much like a tick mark on the y axis):

Notice that this record tells us a lot about what happened to the function on this particular day. (We will follow the events it encountered by working our way towards the positive x axis, starting from the left. That, by convention is always the way we are meant to read off the record of what happened. It is assumed that the valus of x is always increasing.)

Getting back to our story, our function started out with positive slope (somewhere along the negative x axis). This tells us that it must have grown for some time. (Notice how the tangent lines are pointing towards increasing y values when their slope is positive. That's what we mean when we say that the function must have been growing). Eventually, it flattened out, and precisely at the point where the slope of its tangent was zero, the function had a little "high". (We will call that place a local maximum.) After that, our function had a negative slope. This tells us it was decreasing. After some time, the slope flattened out to zero and the function had a local minimum. After that it just grew and grew. (In fact, its growth got faster and faster, since the slope of its tangents got more and more steep - but we need not worry about the quantitative aspect here, since we are trying to convey that the qualitative information about the signs of the slopes is already very revealing.)

We can summarize our findings as follows:

  • A positive derivative means that the function is increasing
  • A negative derivative means that the function is decreasing
  • A zero derivative means that the function has some special behaviour at the given point. It may have a local maximum, a local minimum, (or in some cases, as we will see later, a "turning" point)

As a last remark we should remember that the derivative of a function is, itself, a function since it varies from point to point. If we want to, we could plot it on its own set of axes. You can compare the signs and slopes of the individual tangent lines of the original curve with the graph of the derivative.