Visual Introduction to the Definition of the Derivative

One of the new teachers I will be working with this year will be teaching Calculus, and this has got me thinking about some of the more successful strategies that I used the last time that I taught this course. A bunch of ideas come to mind, but one that sticks out from the others is the visual intro to definition of derivative that I stumbled upon the second year that I taught Calculus.  The first time I taught the course students struggled with this concept.  Using dynamic geometry software made all the difference for helping students learn this concept the next time I taught it.

To demonstrate, say you want to use the defintion of derviative to find the value of the derivative of f(x)=x^2 at x=-2.  Use the diagram below:

I've added a couple of objects to my diagram to help out with this explanation.  First, we note that we are really looking to find the slope of the tangent line of the function f(x)=x^2 at x=-2.  I don't tell students that my software can simply graph this tangent line and give me the slope, nor do I tell them that there are formulas to find the derivative of a function that we can use to find the slope of the tangent line.  That all comes in time.  For now my fixed point, a moving point attached to f(x), and a secant line between these points is all we need.

The next step is a reminder of what a tangent line is.  I let students discuss with a partner and then tell the class what we are looking at.  If the diagram above with a secant line is provided, I can ask students what to do to the moving point to make my dashed line become a tangent line.  I start dragging the moving point towards the fixed points and ask students to tell me when to stop.  If they tell me to stop too soon, I ask them if we can move the blue point closer to the fixed point.  We are visually developing the idea of using a limit to find the slope of a tangent line by physically moving one point as close to another point as possible.  Eventually we put the moving point on the fixed point.

Students notice right away that the secant line is no longer present.  Since you can't turn a secant line into a tangent line using a fixed point and moving point, we will do the best that we can.  And the best that we can do is to move the moving point as close to the fixed point as possible, and use these coordinates to estimate the slope of the tangent line.  I ask students to write this symbolically.  We introduce the following notation:

Fixed Point: (a, f(a))
Moving Point: (x, f(x))

Then the slope of the secant line is

We practice finding the value of the derivative by dragging the moving point as close to the fixed point as possible, and then finding the slope of the secant line.  When we introduce the definition of the derivative of a function at a given value, we use similar language.  Students can tell you the following about the slope of the tangent line with little help:

The slope of the tangent line can be found by taking the limit of the slope of the secant line as the moving point approaches the fixed point.  When we have had this conversation, it is not such a big leap to 

With the visual introduction, students can interpret the parts of this formula and feel empowered when the formula yields the same slope as our estimations did. 

My school district has open enrollment for AP and honors courses, and because of this the Calculus AB classes are very heterogeneous.  There is a nice split between students that have taken all honors math courses, students that are choosing AP Calculus AB as their first honors math course, and students that have never taken a honors/AP course in any subject.  Because of this I chose to keep the functions simple when we were finding derivatives using the definition.  This allowed for a greater focus on the conceptual understanding of derivative, and I believe that the visual introduction using dynamic geometry software and the extra conversations around the topic helped students ace this topic on the following test.  

You can use this premade presentation to introduce the definition of derivative in your own class.  This tool allows you to enter a function as well as coordinates for your fixed point and moving point.  A smaller version of the tool is below, and you can try this out by dragging the moving point.  

You can also use this presentation when talking about when a function is not differentiable, because it is easy to see that the slope of the tangent line will not approach the same value on either side of the cusp.