Reminder: this series is NOT for the student who is attempting to learn calculus for the first time.

**Derivatives** This is dealing with differentiable functions and no, I will NOT be talking about maps between tangent bundles. Yes, my differential geometry and differential topology courses were on the order of 30 years ago or so. đź™‚

In calculus 1, we typically use the following definitions for the derivative of a function at a point: . This is opposed to the *derivative function* which can be thought of as the one dimensional gradient of .

The first definition is easier to use for some calculations, say, calculating the derivative of at a point. (hint, if you need one: use then it is easier to factor). It can be used for proving a special case of the chain rule as well (the case there we are evaluating at and for at most a finite number of points near .)

When introducing this concept, the binomial expansion theorem is very handy to use for many of the calculations.

Now there is another definition for the derivative that is helpful when proving the chain rule (sans restrictions).

Note that as we have . We can now view as a function of which goes to zero as does.

That is, where and is the best linear approximation for at .

We’ll talk about the chain rule a bit later.

But what about the derivative and examples?

It is common to develop intuition for the derivative as applied to nice, smooth..ok, analytic functions. And this might be a fine thing to do for beginning calculus students. But future math majors might benefit from being exposed to just a bit more so I’ll give some examples.

Now, of course, being differentiable at a point means being continuous there (the limit of the numerator of the difference quotient must go to zero for the derivative to exist). And we all know examples of a function being continuous at a point but not being differentiable there. Examples: are all continuous at zero but none are differentiable there; these give examples of a corner, vertical tangent and a cusp respectively.

But for many of the piecewise defined examples, say, for and for the derivative fails to exist because the respective derivative functions fail to be continuous at ; the same is true of the other stated examples.

And of course, we can show that has continuous derivatives at the origin but not derivatives.

**But what about a function with a discontinuous derivative?** Try for and zero at . It is easy to see that the derivative exists for all but the first derivative fails to be continuous at the origin.

The derivative is at and for which is not continuous at the origin.

**Ok, what about a function that is differentiable at a single point only?** There are different constructions, but if for rational, for irrational is both continuous and, yes, differentiable at (nice application of the Squeeze Theorem on the difference quotient).

Yes, there are everywhere continuous, nowhere differentiable functions.