The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook MM Def Deriv.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. See How to manipulate the diagrams for more information on what you can do with them. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

Learning about the derivative as a concept

The derivative $f'(x)$ of $f(x)$ is the function whose value at $a$ is the slope of the line tangent to the graph $y=f(x)$ at the point $(a,f(a))$.

To gain understanding of the concept of derivative the student need to see and play with the pictures that illustrate the definition. This can be done in stages:

Give an intuitive, pictorial explanation of the tangent line.
Show in pictures what the slope of a line is.
Show in pictures how you can approximate the tangent line with secant lines.

Of course, many teachers and textbooks do this. I propose that:

The student will benefit in the long run by spending a whole class session on the intuitive ideas I just described and doing a set homework based only on intuition. Then you can start doing the algebraic stuff.

This post provides some ideas about manipulable diagrams that students can play with to gain intuition about derivatives. Others are possible. There are many on the Mathematica Demonstrations website. There are others written in Java and other languages, but I don't know of a site that tries to collect them in one place.

My claim that the student will benefit in the long run is not something I can verify, since I no longer teach.

Present the tangent line conceptually

The tangent line to a curve

is a straight line that touches the curve at a point on the curve,
and it goes in the same direction that the curve is going, like the red line in the picture below. (See How to manipulate the diagrams.)

My recommendation is that you let the students bring up some of the fine points.

The graph of $y=x^3-x$ has places where the tangent line cuts the curve at another point without being parallel to the curve there. Move the slider to find these places.
The graph of $y=\cos(\pi x)$ has places where the same line is tangent at more than one point on the curve. (This may requre stepping the slider using the incrementers.)
Instigate a conversation about the tangent line to a given straight line.
My post Tangents has other demos intended to bother the students.
Show the unit circle with some tangent lines and make them stare at it until they notice something peculiar.
"This graph shows the tangent line but how do you calculate it?" You can point out that if you draw the curve carefully and then slide a ruler around it so that it is tangent at the point you are interested in, then you can draw the tangent carefully and measure the rise and run with the ruler. This is a perfectly legitimate way to estimate the value of the slope there.

Slope of the tangent line conceptually

This diagram shows the slope of the tangent line as height over width.

Slide the $x$ slider back and forth. The width does not change. The height is measured from the tangent line to the corner, so the height does change; in particular, it changes sign appropriately.
This shows that the standard formula for the derivative of the curve gives the same value as the calculated slope of the tangent. (If you are careful you can find a place where the last decimal places differ.) You may want to omit the "derivative value" info line, but most students in college calculus already know how to calculate the formulas for the derivative of a polynomial– or you can just tell them what it is in this case and promise to show how to calculate the formula later.
Changing the width while leaving $x$ fixed does not change the slope of the tangent line (up to roundoff error).
In fact I could add another parameter that allows you to calculate height over width at other places on the tangent line. But that is probably excessive. (You could do that in a separate demo that shows that basic property that the slope of a straight line does not change depending on where you measure it — that is what a curve being a straight line means.)
This graph provides a way to estimate the slope, but does not suggest a way to come up with a formula for the slope, in other words, a formula for the derivative.

Conceptual calculation of the slope

This diagram shows how to calculate the value of the slope at a point using secant lines to approximate the tangent line. If you have a formula for the function, you can calculate the limit of the slope of the secant line and get a formula for the derivative.

The function $f(x)=x^3-x$.
The secant points are $(x-h,f(x-h))$ and $(x+h, f(x+h))$. $h$ is called "width" in the diagram.
Moving $x$ with the slider shows how the tangent line and secant line have similar slopes.
Moving the width to the left, to $0$ (almost), makes the secant line coincide with the tangent line. So intuitively the limit of the slope of the secant line is the slope of the tangent line.
The distance between the secant points is the Euclidean distance. (It may be that including this information does not help, so maybe it should be left out.)
The slope of the secant line is $\frac{f(x+h)-f(x-h)}{(x+h)-(x-h)}$ when $h\neq0$. This simplifies to $3x^2+h^2-1$, so the limit when $h\to0$ is $3x^2-1$, which is therefore a formula for the derivative function.

Testing intuitive concepts

Most of the work students do when studying derivatives is to solve some word problems (rate of change, maximization) in which the student is expected to come up with an appropriate function $f(x)$ and then know or find out the formula for $f'(x)$ in the process of solving the problem. In other words there is a heavy emphasis on computation and much less on concept.

The student in the past has had to do very few homework problems that test for understanding the concept. Lately some texts do have problems that test the concept, for example:

This is the graph of a function and its derivative. Which one is the function and which is its derivative?

Note that the problem does not give you the formula for the function, nor does it have to.

Many variations are possible, all involving calculating parameters directly from the graph:

"These are the first and second derivatives of a function. Where (within the bounds of the graph) is the function concave up?"
"These are the first and second derivatives of a function. Where (within the bounds of the graph) are its maxima and minima?"
"This straight line is the derivative of a function. Show that the function is a quadratic function and measure the slope of the line in order to estimate some of the coefficients of the quadratic."

How to manipulate the diagrams

You can move the sliders back and forth to to move to different points on the curve.
In the first diagram, you can click on one of the four buttons to see how it works for various curves.
The arrow at the upper right makes it run automatically in a not very useful sort of way.
The little plus sign below the arrow opens up some other controls and a box showing the value of $a$, including step by step operation (plus and minus signs).
If you are using Mathematica, you can enter values into the box, but if you are using CDF Player, you can only manipulate the number using the slider or the plus and minus incrementers.

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The interactive examples in this post require installing Wolfram CDF Player., which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook Tangent Line.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

The diagram above shows you the tangent line to the curve $y=x^3-x$ at a specific point. The slider allows you to move the point around, and the tangent line moves with it. You can click on one of the plus signs for options about things you can do with the slider. (Note: This is not new. Many other people have produced diagrams like this one.)

I have some comments to make about this very simple diagram. I hope they raise your consciousness about what is going on when you use a manipulable demonstration.

Farming out your abstraction load

A diagram showing a tangent line drawn on the board or in a paper book requires you visualize how the tangent line would look at other points. This imposes a burden of visualization on you. Even if you are a new student you won't find that terribly hard (am I wrong?) but you might miss some things at first:

There are places where the tangent line is horizontal.
There are places where some of the tangent lines cross the curve at another point. Many calculus students believe in the myth that the tangent line crosses the curve at only one point. (It is not really a myth, it is a lie. Any decent myth contains illuminating stories and metaphors.)
You may not envision (until you have some experience anyway) how when you move the tangent line around it sort of rocks like a seesaw.

You see these things immediately when you manipulate the slider.

Manipulating the slider reduces the load of abstract thinking in your learning process. You have less to keep in your memory; some of the abstract thinking is offloaded onto the diagram. This could be described as contracting out (from your head to the picture) part of the visualization process. (Visualizing something in your head is a form of abstraction.)

Of course, reading and writing does that, too. And even a static graph of a function lowers your visualization load. What interactive diagrams give the student is a new tool for offloading abstraction.

You can also think of it as providing external chunking. (I'll have to think about that more…)

Simple manipulative diagrams vs. complicated ones

The diagram above is very simple with no bells and whistles. People have come up with much more complicated diagrams to illustrate a mathematical point. Such diagrams:

May give you buttons that give you a choice of several curves that show the tangent line.
May give a numerical table that shows things like the slope or intercept of the current tangent line.
May also show the graph of the derivative, enabling you to see that it is in fact giving the value of the slope.

Such complicated diagrams are better suited for the student to play with at home, or to play with in class with a partner (much better than doing it by yourself). When the teacher first explains a concept, the diagrams ought to be simple.

Examples

The Definition of derivative demo (from the Wolfram Demonstration Project) is an example that provides a table that shows the current values of some parameters that depend on the position of the slider.
The Wolfram demo Graphs of Taylor Polynomials is a good example of a demo to take home and experiment extensively with. It gives buttons to choose different functions, a slider to choose the expansion point, another one to choose the number of Taylor polynomials, and other things.
On the other hand, the Wolfram demo Tangent to a Curve is very simple and differs from the one above in one respect: It shows only a finite piece of the tangent line. That actually has a very different philosophical basis: it is representing for you the stalk of the tangent space at that point (the infinitesimal vector that contains the essence of the tangent line).
Brian Hayes wrote an article in American Scientist containing a moving graph (it moves only on the website, not in the paper version!) that shows the changes of the population of the world by bars representing age groups. This makes it much easier to visualize what happens over time. Each age group moves up the graph — and shrinks until it disappears around age 100 — step by step. If you have only the printed version, you have to imagine that happening. The printed version requires more abstract visualization than the moving version.
Evaluating an algebraic expression requires seeing the abstract structure of the expression, which can be shown as a tree. I would expect that if the students could automatically generate the tree (as you can in Mathematica) they would retain the picture when working with an expression. In my post computable algebraic expressions in tree form I show how you could turn the tree into an evaluation aid. See also my post Syntax trees.

This blog has a category "Mathematica" which contains all the graphs (many of the interactive) that are designed as an aid to offloading abstraction.

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