Chapter 2:

Derivative Graphs, Trigonometric Derivatives, Properties of Derivatives:

The derivative FUNCTION, f ' (x).

Given a particular function, f(x), the graph of its derivative function, f ' (x) can be thought of as a table of values of the slopes of the tangent lines to f(x) throughout its domain.  Consequently, when f(x) has positive tangent slopes, f ' (x) is positive, when f(x) has negative tangent slopes, f ' (x) is negative, and when f(x) has horizontal tangents, f ' (x) has roots (zeros).  Look carefully at the following GeoGebra applet:

Applet:

Notice that when f(x) is DECREASING, f ' (x) is NEGATIVE, and when f(x) is INCREASING, f ' (x) is POSITIVE.  Notice, also, that it is quite possible for f '(x) to be increasing, while f(x) is decreasing; it is similarly possible for f ' (x) to be negative while f(x) is positive.  So it's important to note both what the derivative graph does say, and does not say, about the original function graph.

Look at the next function, and trace out its derivative.  Notice that f(x) is continuous throughout.  What about f ' (x) ?

Applet:

So, it is possible to have continuous functions, for which the derivative is not defined at certain places.  What about the converse?  That is, suppose you know that the derivative of a function exists at a particular point.  Must the function be continuous there?  The answer is yes:

Theorem:  If f(x) is a function, and f ' (c) exists at x = c, then f(x) must be continuous at x = c.

Proof:

So, the set of "differentiable functions" is a subset of the set of "continuous functions."

Amazingly, there exist continuous functions that are "nowhere differentiable," that is, functions which are continuous at every point, but also have "corners" at every point.  Needless to say, they're impossible to sketch, and they are also rather difficult to define, but here's an example.

Here are a few simpler examples (they're still a bit dense -- see if you can follow along, but pay special attention to the pictures.)

Example #1:  f(x) = x^2/3

Applet:

Example #2:  f(x) = | x² - 4 |

Applet:

Again, go over the notes, above, but at this point, you should be more concerned that you understand the pictures and applets that demonstrate the situation.

Next: The Trigonometric Derivatives: