Summary and suggestions for further study

So what is the derivative, after all?

We have discussed the notions of the derivative in many forms and guises on these pages. Perhaps it is time for a summary of all these forms, and a simple statement of what, after all, the derivative "really is". First, let us review the many ways in which the idea of a derivative can be represented:

The derivative

The derivative measures the steepness of the graph of a function at some particular point on the graph. Thus, the derivative is a slope. (That means that it is a ratio of change in the value of the function to change in the independent variable.)
If the independent variable happens to be "time", we often think of this ratio as a rate of change (an example is velocity)
If we zoom in on the graph of the function at some point so that the function looks almost like a straight line, the derivative at that point is the slope of the line. This is the same as saying that the derivative is the slope of the tangent line to the graph of the function at the given point.
The slope of a secant line (line connecting two points on a graph) approaches the derivative when the interval between the points shrinks down to zero.
The derivative is also, itself, a function: it varies from place to place. For example, the velocity of a car may change from moment to moment as the driver speeds up or slows down.

The last remark is quite important and interesting: it tells us that when we have finished determining the derivative of some particular function everywhere, we get another function! We could then talk about its derivative ! (Ofcourse, we do this very often without realizing it! Whenever we talk about acceleration we are talking about the derivative of a derivative, i.e. the rate of change of a velocity.) Second derivatives (and third derivatives, and so on) are also functions ! Each one tells us about the rate of change of the previous function in this pyramid scheme.

We have used a lot of words to try to describe what the derivative is. Mathematicians try to avoid lots of words, aiming at precision and succinctness. Let's take a look at what they might do instead.

A mathematician's code

Mathematicians have developed a kind of "secret code" that says all of the things we have enumerated above with a few strokes of the pen. It actually took centuries to develop this code to the point where it became part of the mathematical society's accepted language, but now it is used universally. There is nothing particularly mysterious, interesting, or important about the details of the code itself (which "they" call "mathematical notation") but because everyone uses it, we should at least tell you about it.

A mathematician would start like this:

Definition of the derivative

Assume that $ y=f(x) $ is a differentiable function at the point $ x_0 $ . Then the derivative of the function at this point is:

$\frac {dy}{dx}|_{x_0} = f'(x_0) = \lim_{h \to 0} \frac{ f(x_0+h)-f(x_0)}{h}$

Let's work at "cracking this code" piece by piece. It may help to look at the diagram below while we are doing this:

First, notice that there is some "fine print" attached: "Assume that $ y=f(x) $ is a differentiable function." We have hinted at the fact that not all functions have a derivative at every point. If the graph of the function has a sharp corner or kink at some point (for example, the absolute value function $ f(x) = |x| $ has a sharp graph at the point $ x=0 $ ), or if its graph has an "infinite slope" near some point, then we should not expect a derivative to be defined there.

Next, the expression on the left hand side with the vertical bar

$\frac {dy}{dx}|_{x_0}$

just means "the derivative of the function at ". The expression $ f'(x_0) $ means exactly the same thing.

Now let's look at the right hand side:

First, notice that, as expected, there is a ratio which looks like

$\frac{ f(x_0+h)-f(x_0)}{h}$

The top represents a change in the value of the function between the two points whose x values are, $ x_0 $ and $ x_0+h $ . The change in the value of the function is shown on our diagram with the green line.

The bottom of the ratio is the change in the x value itself ! $ (x_0+h)-x_0= h $ . This change in x value is shown with a magenta colored line in the diagram.

This means that this ratio is in fact, a slope. It is the slope of the secant line connecting the two points on the graph: $(x_0, f(x_0)) ~~{\rm and}~~(x_0+h, f(x_0+h))$ . So far, we are in agreement with the verbal descriptions of the derivative.

The final bit we need to decipher is: $\lim_{h \to 0}$ . This says, in words "the limit as $ h $ approaches zero", which, even more simply worded is: the value that the ratio approaches as the two points get closer and closer to one another. This is precisely what we were saying when we talked about the way that the secant line approaches the tangent line on the graph of a function.

For your consideration:

(1) What do you think might happen if we tried to calulate the derivative of the absolute value function, at the point ? Hint: what is the slope of this function along the positive x axis? along the negative x axis? So what should we say is the "slope" at zero?
(2) Can you think of any functions that we have seen so far that have an "infinite slope" at some point ? Hint: look carefully at the page with fractional powers of x.
(3) So far we have looked mainly at functions that are fairly smooth, and have no breaks in their graphs. But sometimes we are interested in the so called discontinuous functions, whose graphs have a break in them. For example, when you turn on a light switch, the current in the circuit jumps abruptly from zero to some other value. What do you think would happen to the derivative of such a function at the point at which there is a break in its graph?
(4) Review the calculation of the velocity of a falling ball. Show that we have basically computed the ratio shown above and let the quantity get smaller and smaller in magnitude.