$$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\FF}{\mathbb{F}}
\renewcommand{\epsilon}{\varepsilon}
% ALTERNATE VERSIONS
% \newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}}
% \newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}}
% \newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}}
% \newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}}
% \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}}
\newcommand{\uppersum}[1]{U_{#1}}
\newcommand{\lowersum}[1]{L_{#1}}
\newcommand{\upperint}[1]{U_{#1}}
\newcommand{\lowerint}[1]{L_{#1}}
\newcommand{\rsum}[1]{{\textstyle\sum_{#1}}}
% extra auxiliary and additional topic/proof
\newcommand{\extopic}{\bigstar}
\newcommand{\auxtopic}{\blacklozenge}
\newcommand{\additional}{\oplus}
\newcommand{\partitions}[1]{\mathcal{P}_{#1}}
\newcommand{\sampleset}[1]{\mathcal{S}_{#1}}
\newcommand{\erf}{\operatorname{erf}}
$$
In 22 Definition we define the derivative as a limit of difference quotients
In 23 Working with Derivatives we investigate the basic properties of differentiable functions and their arithmetic
In 24 Differentiable Functions we prove the Mean Value Theorem, a cornerstone result in real analysis
In 25 Applications we study a wide range of applications of the mean value theorem, from understanding maxima and minima to L’Hospitals rule.
In 26 Power Series we investigate the differentiability of power series
In 27 Elementary Functions we use calculus to single out the natural exponential, natural logarithm, and natural units (radians) for the trigonometric functions.