Definition: Derivative, Differentiable Functions

Let $D\subseteq\mathbb R$ ($D$ is a subset of real numbers) and let $f:D\to\mathbb R$ be a function. Using the concept of the difference quotient,

\[\Delta f(y,x)=\frac {f(y)-f(x)}{y-x},\quad y\neq x,\]

for a fixed $x\in D$ and all $\xi\in D$ with $\xi\neq x$, we define a function $g:D\to\mathbb R$ by

\[g(\xi):=\Delta f(\xi,x)=\frac {f(\xi)-f(x)}{\xi-x},\xi\neq x.\]

If $g$ is continuous at $x$, its limit at $x$ exists¹ and is unique². This limit defines a new function

\[f'(x):=\lim_{\substack{\xi\to x\\\xi\in D\setminus\{x\}}} g(\xi)=\lim_{\substack{\xi\to x\\\xi\in D\setminus\{x\}}}\frac {f(\xi)-f(x)}{\xi-x}.\]

called the derivative of $f$ at $x$.

By a change of variables, the derivative $f'(x)$ can also be defined as \[f'(x):=\lim_{h\to 0}\frac {f(x+h)-f(x)}{h}.\] In this definition, only those sequences $(h_n)_{n\in\mathbb N}$ with $\lim h_n=0$ are allowed, for which $h_n\neq 0$ and $x+h_n\in D$.
Instead of writing $f'(x)$, the derivative can also be written³ as $\frac {d f(x)}{dx}$.
$f$ has a derivative at $x$ $\Longleftrightarrow f$ is differentiable at $x$.
$f$ has a derivative at every $x\in D$ $\Longleftrightarrow f$ is differentiable on $D$.

Corollaries: 1 2 3 4 5
Definitions: 6 7 8 9 10
Examples: 11
Lemmas: 12
Parts: 13
Proofs: 14 15 16 17 18 19 20 21
Propositions: 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
Sections: 45 46
Theorems: 47 48 49

Thank you to the contributors under CC BY-SA 4.0!

Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983

Please note that the continuity of $g$ at $x$ requires the existence of at least one sequence $(x_n)_{n\in\mathbb N}$, $x_n\in D$ with $\lim_{n\to\infty} x_n=x$. ↩
The uniqueness of the limit means that for every sequence $(\xi_n)_{n\to\infty}$, $\xi_n\in D\setminus\{x\}$ with $\lim_{n\to\infty} \xi_n=x$ we have $g(\xi_n)=f'(x)$. ↩
The notation $\frac {d f(x)}{dx}$ can, in comparison with the notation $f'(x)$, sometimes become cumbersome. For instance, if $x=0$, we can write $f'(0)$, but we cannot write $\frac {d f(0)}{d0}$. In this case it is necessary to write $\frac {d f}{dx}(0)$ or ${\frac {d f(x)}{dx}}|^{d0}_{x=0}$ ↩