Chapter 2:

Slopes, average rates of change, limits and instantaneous rates of change:

Product Rule, Quotient Rule, Examples and Demonstrations:

Notice that the "linearity" properties of derivatives are easy to see when you're looking at the graphs:

First, examine what happens to a constant multiple of a function:

Applet:

It is clear that D_x[K*f(x)] = K*f ' (x).

Next, think about what happens to the slopes of tangents when you add two functions together:

Applet:

So, the Linearity properties we looked at last time are also apparent when you look at the graphs.

There's a third property which seems sort of trivial, but turns out to be very important, especially as we look at anti-derivatives in Chapter 4.  It's really a special case of the above example, but we usually think of it as a separate notion. 

If you take a function, f(x) and simply displace it vertically (add or subtract a constant), you don't change the slopes of the tangent lines at all.  In fact, for any given "A," the tangent lines are parallel; you can think of the displaced function, g(x) = f(x) + k as parallel to f(x):

Applet:

Now, we get to a new, very important, and somewhat complicated rule.  It's probably as important to know what's NOT true in this case, as it is to know what IS true.  What happens to the slope of the resulting function, when you multiply two functions together? Notice that this is not a constant times a function (like at the top of the page), but a function times a different function.  Take a look at the applet below, and notice, first, that you clearly can't just multiply slopes together.  In the upper left, you'll see what the actual relationship is, and we'll prove it shortly.

Applet:

So, what's going on?  Well, let's go back to the definition of the derivative, and see if we can find some relationship between the slope of the product of two functions, and the original two functions:

In operator notation, we have:

(Many books, including ours, write the sum the other way.  I prefer it this way, since it's the same order you'll need for the Quotient Rule, below.  "Always start with the derivative of f," seems easier to remember, for me, anyway.)

Examples:

Next: The Quotient Rule: