# Real functions/Differentiability/Introduction/Section

In this section, we consider functions

${\displaystyle f\colon D\longrightarrow \mathbb {R} ,}$

where ${\displaystyle {}D\subseteq \mathbb {R} }$ is a subset of the real numbers. We want to explain what it means that such a function is differentiable in a point ${\displaystyle {}a\in D}$. The intuitive idea is to look at another point ${\displaystyle {}x\in D}$, and to consider the secant, given by the two points ${\displaystyle {}(a,f(a))}$ and ${\displaystyle {}(x,f(x))}$, and then to let "${\displaystyle {}x}$ move towards ${\displaystyle {}a}$“. If this limiting process makes sense, the secants tend to become a tangent. However, this process only has a precise basis, if we use the concept of the limit of a function as defined earlier.

## Definition

Let ${\displaystyle {}D\subseteq \mathbb {R} }$ be a subset, ${\displaystyle {}a\in D}$ a point, and

${\displaystyle f\colon D\longrightarrow \mathbb {R} }$

a function. For ${\displaystyle {}x\in D}$, ${\displaystyle {}x\neq a}$, the number

${\displaystyle {\frac {f(x)-f(a)}{x-a}}}$

is called the difference quotient of ${\displaystyle {}f}$ for

${\displaystyle {}a}$ and ${\displaystyle {}x}$.

The difference quotient is the slope of the secant at the graph, running through the two points ${\displaystyle {}(a,f(a))}$ and ${\displaystyle {}(x,f(x))}$. For ${\displaystyle {}x=a}$, this quotient is not defined. However, a useful limit might exist for ${\displaystyle {}x\rightarrow a}$. This limit represents, in the case of existence, the slope of the tangent for ${\displaystyle {}f}$ in the point ${\displaystyle {}(a,f(a))}$.

## Definition

Let ${\displaystyle {}D\subseteq \mathbb {R} }$ be a subset, ${\displaystyle {}a\in D}$ a point, and

${\displaystyle f\colon D\longrightarrow \mathbb {R} }$

a function. We say that ${\displaystyle {}f}$ is differentiable in ${\displaystyle {}a}$ if the limit

${\displaystyle \operatorname {lim} _{x\in D\setminus \{a\},\,x\rightarrow a}\,{\frac {f(x)-f(a)}{x-a}}}$

exists. In the case of existence, this limit is called the derivative of ${\displaystyle {}f}$ in ${\displaystyle {}a}$, written

${\displaystyle f'(a).}$

The derivative in a point ${\displaystyle {}a}$ is, if it exists, an element in ${\displaystyle {}\mathbb {R} }$. Quite often one takes the difference ${\displaystyle {}h:=x-a}$ as the parameter for this limiting process, that is, one considers

${\displaystyle \operatorname {lim} _{h\rightarrow 0}\,{\frac {f(a+h)-f(a)}{h}}.}$

The condition ${\displaystyle {}x\in D\setminus \{a\}}$ translates then to ${\displaystyle {}a+h\in D}$, ${\displaystyle {}h\neq 0}$. If the Function ${\displaystyle {}f}$ describes a one-dimensional movement, meaning a time-dependent process on the real line, then the difference quotient ${\displaystyle {}{\frac {f(x)-f(a)}{x-a}}}$ is the average velocity between the (time) points ${\displaystyle {}a}$ and ${\displaystyle {}x}$ and ${\displaystyle {}f'(a)}$ is the instantaneous velocity in ${\displaystyle {}a}$.

## Example

Let ${\displaystyle {}s,c\in \mathbb {R} }$, and let

${\displaystyle \alpha \colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto sx+c,}$

be an affine-linear function. To determine the derivative in a point ${\displaystyle {}a\in \mathbb {R} }$, we consider the difference quotient

${\displaystyle {}{\frac {(sx+c)-(sa+c)}{x-a}}={\frac {sx-sa}{x-a}}=s\,.}$

This is constant and equals ${\displaystyle {}s}$, so that the limit of the difference quotient as ${\displaystyle {}x}$ tends to ${\displaystyle {}a}$ exists and equals ${\displaystyle {}s}$ as well. Hence, the derivative exists in every point and is just ${\displaystyle {}s}$. The slope of the affine-linear function is also its derivative.

## Example

We consider the function

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2}.}$

The difference quotient for ${\displaystyle {}a}$ and ${\displaystyle {}a+h}$ is

${\displaystyle {}{\frac {f(a+h)-f(a)}{h}}={\frac {(a+h)^{2}-a^{2}}{h}}={\frac {a^{2}+2ah+h^{2}-a^{2}}{h}}={\frac {2ah+h^{2}}{h}}=2a+h\,.}$

The limit of this, as ${\displaystyle {}h}$ tends to ${\displaystyle {}0}$, is ${\displaystyle {}2a}$. The derivative of ${\displaystyle {}f}$ in ${\displaystyle {}a}$ is therefore ${\displaystyle {}f'(a)=2a}$.