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 \(x\) exists^{1} and is unique^{2}. 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 \(x\).
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
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 \({\frac {d f(x)}{dx}}|^{d0}_{x=0}\) ↩