Derivative Patterns

In Section 1.2 we saw that the "holey" function $\mathrm{h}(\mathrm{x})= \begin{cases}2 & \text { if } \mathrm{x} \text { is a rational number } \\ 1 & \text { if } \mathrm{x} \text { is an irrational number }\end{cases}$

is discontinuous at every value of $x$, so at every $x$  $h(x)$ is not differentiable. We can create graphs of continuous functions that are not differentiable at several places just by putting corners at those places, but how many corners can a continuous function have? How badly can a continuous function fail to be differentiable? 

In the mid–1800s, the German mathematician Karl Weierstrass surprised and even shocked the mathematical world by creating a function which was continuous everywhere but differentiable nowhere - a function whose graph was everywhere connected and everywhere bent! He used techniques we have not investigated yet, but we can start to see how such a function could be built. 

Start with a function $f_1$ (Fig. 4) which zigzags between the values $+1/2$ and $–1/2$ and has a "corner" at each integer. This starting function $f_1$ is continuous everywhere and is differentiable everywhere except at the integers. Next create a list of functions $f_{2}, f_{3}, f_{4}, \ldots$, each of which is a lot shorter but with many more "corners" than the previous ones. For example, we might make $\mathrm{f}_{2}$ zigzag between the values $+1 / 4$ and $-1 / 4$ and have "corners" at $\pm 1 / 2, \pm 3 / 2$, $\pm 5 / 2$, etc., and $\mathrm{f}_{3}$ zigzag between $+1 / 9$ and $-1 / 9$ and have "corners" at $\pm 1 / 3$, $\pm 2 / 3, \pm 4 / 3$, etc. If we add $\mathrm{f}_{1}$ and $\mathrm{f}_{2}$, we get a continuous function (since the sum of two continuous functions is continuous) which will have corners at $0, \pm 1 / 2, \pm 1$, $\pm 3 / 2, \ldots$ If we then add $f_{3}$ to the previous sum, we get a new continuous function with even more corners. If we continue adding the functions in our list "indefinitely", the final result will be a continuous function which is differentiable nowhere.

We haven't developed enough mathematics here to precisely describe what it means to add an infinite number of functions together or to verify that the resulting function is nowhere differentiable, but we will. You can at least start to imagine what a strange, totally "bent" function it must be. 

Until Weierstrass created his "everywhere continuous, nowhere differentiable" function, most mathematicians thought a continuous function could only be "bad" in a few places, and Weierstrass' function was (and is) considered "pathological", a great example of how bad something can be. The mathematician Hermite expressed a reaction shared by many when they first encounter Weierstrass' function: 

"I turn away with fright and horror from this lamentable evil of functions which do not have derivatives".

IMPORTANT RESULTS

Power Rule For Functions: $\quad D\left(f^{n}(x)\right)=n \cdot f^{n-1}(x) \cdot D(f(x))$

Derivatives of the Trigonometric Functions:

$\begin{align*} \begin{array}{lll} \mathbf{D}(\sin (\mathrm{x}))=\cos (\mathrm{x}) & \mathrm{D}(\tan (\mathrm{x}))=\sec ^{2}(\mathrm{x}) & \mathrm{D}(\sec (\mathrm{x}))=\sec (\mathrm{x}) \tan (\mathrm{x}) \\ \mathbf{D}(\cos (\mathrm{x}))=-\sin (\mathrm{x}) & \mathrm{D}(\cot (\mathrm{x}))=-\csc ^{2}(\mathrm{x}) & \mathrm{D}(\csc (\mathrm{x}))=-\csc (\mathrm{x}) \cot (\mathrm{x}) \end{array} \end{align*}$

Derivatives of the Exponential Function: $\quad D\left(e^{x}\right)=e^{x}$