Real functions/Differentiability/Introduction/Section

In this section, we consider functions

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

where ${}D\subseteq \mathbb {R}$ is a subset of the real numbers. We want to explain what it means that such a function is differentiable in a point ${}a\in D$. The intuitive idea is to look at another point ${}x\in D$, and to consider the secant, given by the two points ${}(a,f(a))$ and ${}(x,f(x))$, and then to let "${}x$ move towards ${}a$“. 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 ${}D\subseteq \mathbb {R}$ be a subset, ${}a\in D$ a point, and

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

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

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

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

${}a$ and ${}x$.

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

Definition

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

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

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

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

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

$f'(a).$

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

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

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

Example

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

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

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

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

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

Example

We consider the function

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

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

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

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