#### Chapter 7 Differential Calculus

Section 7.1 Derivative of a Function

# 7.1.3 Derivative

##### Notation of the Derivative 7.1.3
In mathematics, sciences and engineering, different but equivalent notations for derivatives are used:

$f\text{'}\left({x}_{0}\right)=\frac{df}{dx}\left({x}_{0}\right)=\frac{d}{dx}f\left({x}_{0}\right) .$

These different notations all denote the derivative of the function $f$ at the point ${x}_{0}$.

If the derivative is to be calculated using the difference quotient $\frac{f\left(x\right)-f\left({x}_{0}\right)}{x-{x}_{0}}$, then it is often convenient to rewrite the difference quotient in another way. Denoting the difference of $x$ and ${x}_{0}$ by $h:=x-{x}_{0}$ (see figure below), the difference quotient can be rewritten as

$\frac{f\left(x\right)-f\left({x}_{0}\right)}{x-{x}_{0}}=\frac{f\left({x}_{0}+h\right)-f\left({x}_{0}\right)}{h} ,$

where $x={x}_{0}+h$. There is no statement about whether $x$ has to be greater or less than ${x}_{0}$. Hence, the quantity $h$ can take positive or negative values. To determine the derivative of the function $f$, the limit for $h\to 0$ has to be calculated:

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

If this limit exists for all points ${x}_{0}$ in a function's domain, then the function is said to be differentiable (everywhere). Many of the common functions are differentiable. However, a simple example of a function that is not differentiable everywhere is the absolute value function $f:ℝ\to ℝ$ with $x\to f\left(x\right):=|x|$.
##### Example 7.1.4
The absolute value function (see Module 6, Section 6.2.5) is not differentiable at the point ${x}_{0}=0$. The difference quotient of $f$ at the point ${x}_{0}=0$ is:

$\frac{f\left(0+h\right)-f\left(0\right)}{h}=\frac{|h|-|0|}{h}=\frac{|h|}{h} .$

Since $h$ can be greater or less than $0$, two cases are to be distinguished: For $h>0$, we have $\frac{|h|}{h}=\frac{h}{h}=1$, and for $h<0$, we have $\frac{|h|}{h}=\frac{-h}{h}=-1$. In these two cases, the limiting process, i.e. $h$ approaching $0$, results in two different values ($1$ and $-1$). Thus, the limit of the difference quotient at the point ${x}_{0}=0$ does not exist. Hence, the absolute value function is not differentiable at the point ${x}_{0}=0$.
The graph changes its direction at the point $\left(0;0\right)$ abruptly: Casually speaking, one says that the graph of the function has a kink at the point $\left(0;0\right)$. Likewise, if a function has a jump at a certain point, a unique tangent line to the graph at this point does not exist and thus, the function has no derivative at this point.