4.2 The derivative of a function

Now begins what is ordinarily thought of as the first main subject of calculus, the derivative.

Let $S$ be a subset of $R,$ let $f:S\to C$ be a complex-valued function (of a real variable), and let $c$ be an element of the interior of $S.$ We say that $f$ is differentiable at c if

$$\underset{h\to 0}{lim}\frac{f(c+h)-f\left(c\right)}{h}$$

exists. (Here, the number $h$ is a real number.)

Analogously, let $S$ be a subset of $C,$ let $f:S\to C$ be a complex-valued function (of a complex variable), and let $c$ be an element of the interior of $S.$ We say that $f$ is differentiable at c if

$$\underset{h\to 0}{lim}\frac{f(c+h)-f\left(c\right)}{h}$$

exists. (Here, the number $h$ is a complex number.)

If $f:S\to C$ is a function either of a real variable or a complex variable,and if $S^{\text{'}}$ denotes the subset of $S$ consisting of the points $c$ where $f$ is differentiable, we define a function $f^{\text{'}}:S^{\text{'}}\to C$ by

$$f^{\text{'}}\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}.$$

The function $f^{\text{'}}$ is called the derivative of $f.$

A continuous function $f:[a,b]\to C$ that is differentiable at each point $x\in (a,b),$ and whose the derivative $f^{\text{'}}$ is continuous on $(a,b),$ is called a smooth function on $[a,b].$ If there exists a partition $\{a=x_0<x_1<...<x_n=b\}$ of $[a,b]$ such that $f$ is smooth on each subinterval $[x_{i-1},x_i],$ then $f$ is called piecewise smooth on $[a,b].$

Higher order derivatives are defined inductively. That is, ${f^{\text{'}}}^{\text{'}}$ is the derivative of $f^{\text{'}},$ and so on. We use the symbol $f^{\left(n\right)}$ for the $n$ th derivative of $f.$

REMARK In the definition of the derivative of a function $f,$ we are interested in the limit, as $h$ approaches 0, not of $f$ but of the quotient $q\left(h\right)=\frac{f(c+h)-f\left(c\right)}{h}.$ Notice that 0 is not in the domain of the function $q,$ but 0 is a limit point of that domain. This is the reason whywe had to make such a big deal above out of the limit of a function. The function $q$ is often called the differential quotient.

REMARK As mentioned in [link] , we are often interested in solving for unknowns that are functions.The most common such problem is to solve a differential equation. In such a problem, there is an unknown functionfor which there is some kind of relationship between it and its derivatives. Differential equations can be extremely complicated, and manyare unsolvable. However, we will have to consider certain relatively simple ones in this chapter, e.g., $f^{\text{'}}=f,$ $f^{\text{'}}=-f,$ and ${f^{\text{'}}}^{\text{'}}=\pm f.$

There are various equivalent ways to formulate the definition of differentiable, and each of these ways has its advantages.The next theorem presents one of those alternative ways.

Let $c$ belong to the interior of a set $S$ (either in $R$ or in $C$ ), and let $f:S\to C$ be a function. Then the following are equivalent.

1.   $f$ is differentiable at $c.$ That is,
   $$\underset{h\to 0}{lim}\frac{f(c+h)-f\left(c\right)}{h}\text{exists.}$$
2. $$\underset{x\to c}{lim}\frac{f\left(x\right)-f\left(c\right)}{x-c}\text{exists.}$$
3. There exists a number $L$ and a function $\theta$ such that the following two conditions hold:
   $$f(c+h)-f\left(c\right)=Lh+\theta \left(h\right)$$
   and
   $$\underset{h\to 0}{lim}\frac{\theta \left(h\right)}{h}=0.$$
   In this case, $L$ is unique and equals $f^{\text{'}}\left(c\right),$ and the function $\theta$ is unique and equals $f(c+h)-f\left(c\right)-f^{\text{'}}\left(c\right)h.$

That (1) and (2) are equivalent follows from [link] by writing $x$ as $c+h.$

Suppose next that $f$ is differentiable at $c,$ and define