derivativesOnly.tex

\documentclass[a4paper,12pt]{article}

\usepackage{cmap} % поиск в PDF
\usepackage{mathtext} % русские буквы в формулах
\usepackage[english,russian]{babel} % локализация и переносы
\usepackage[T2A]{fontenc} % кодировка
\usepackage[utf8]{inputenc} % кодировка исходного текста
\usepackage{amsmath}
\usepackage{amsfonts}

\usepackage{multicol}
\usepackage[thinc]{esdiff}
\usepackage{relsize}
\usepackage{graphicx}

\usepackage[margin=0.5in]{geometry}

\pagenumbering{gobble}

\newcommand{\VCenter}[2]{
  \vcenter{\hbox{\scalebox{#1}{$#2$}}}
}

\newcommand{\sep}{\end{multicols}\begin{multicols}}
\newcommand{\Cstart}{\begin{enumerate}\begin{multicols}{2}}
\newcommand{\Csep}{\sep{2}}
\newcommand{\Cend}{\end{multicols}\end{enumerate}}
\newcommand{\Cnk}[2]{\VCenter{1.5}{C_{#1}^{#2}}}
\newcommand{\ds}{\displaystyle}
\newcommand{\Ds}{\ds \vphantom{\sum_n^k}}
\newcommand{\Dfs}{\ds \vphantom{1\over2}}

\newcommand{\Tk}{n} \newcommand{\Tnotk}{k}
% \newcommand{\Tk}{k} \newcommand{\Tnotk}{n}

% \newcommand{\T}{\underset{f(a)}{\operatorname{T}_\Tnotk(x)}}
\newcommand{\T}{\operatorname{T}_\Tnotk(x)}

\newcommand{\Cbin}[2]{\Cnk{#1}{#2}}
% \newcommand{\Cbin}[2]{\binom{#1}{#2}}

% \newcommand{\al}{a}
\newcommand{\al}{\alpha}

\newcommand{\Tn}{\Tnotk} \newcommand{\To}{+\operatorname{o}(x^\Tnotk)}
% \newcommand{\Tn}{\infty} \newcommand{\To}{}

\begin{document}
  \Cstart
  \sep{4}
    \item $\Dfs \ln'(x) = \frac{1}{x}$
    \item $\Dfs (\log_bx)' = \frac{1}{x \ln b}$
    \item $\Dfs (x^\al)' = \al \cdot x^{\al-1}$
    \item $\Dfs (\al^x)' = \al^x \ln \al$
  \sep{4}
    \item $\Ds \sin'(x) = \cos(x)$
    \item $\Ds \cos'(x) = -\sin(x)$
    \item $\Ds \left(1 \over x\right)' = -\frac{1}{x^2}$
    \item $\Ds \left(\sqrt{x}\right)' = \frac{1}{2\sqrt{x}}$
  \Csep
    \item $\ds \arcsin'(x) = \frac{1}{\sqrt{1-x^2}}$
    \item $\ds \arccos'(x) = -\frac{1}{\sqrt{1-x^2}}$
  \Csep
    \item $\ds \arctan'(x) = \frac{1}{x^2+1}$
    \item $\ds \tan'(x) = \frac{1}{\cos^2(x)}$
  \Csep
    \item $\ds \arcctg'(x) = -\frac{1}{x^2+1}$
    \item $\ds \ctg'(x) = -\frac{1}{\sin^2(x)}$
  \end{multicols}
    \vspace{0.1cm} \hrule \vspace{-0.1cm}
  % \begin{multicols}{2}
    % \item $\ds (1+x)^\al = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} = \sum_{\Tk=0}^{\Tn} \Cbin{\al}{\Tk} x^\Tk \To$
    % \item $\ds \ln(1+x) = \sum_{\Tk=1}^{\Tn} \frac{(-1)^{\Tk-1}x^\Tk}{\Tk} \To$
  % \Csep
    % \item $\ds \cos x = \sum_{\Tk=0}^{\Tn} (-1)^\Tk \frac{x^{2\Tk}}{2\Tk!} \To$
    % \item $\ds \sin x = \sum_{\Tk=0}^{\Tn} (-1)^\Tk \frac{x^{2\Tk+1}}{(2\Tk+1)!} \To$
  % \Csep
    % \item $\ds e^x = \sum_{\Tk=0}^{\Tn} \frac{x^\Tk}{\Tk!} \To$
    % \item $\Ds \frac{1}{1-x} = \sum_{\Tk=0}^{\Tn} x^\Tk \To$
  % \end{multicols}
    \item $\ds (1+x)^\al
      = \sum_{\Tk=0}^{\Tn} \Cbin{\al}{\Tk} x^\Tk \To$
    \item $\ds \ln(1+x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \operatorname{o}(x^3)
      = \sum_{\Tk=1}^{\Tn} \frac{(-1)^{\Tk-1}x^\Tk}{\Tk} \To$
    \item $\ds \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \operatorname{o}(x^5)
      = \sum_{\Tk=0}^{\Tn} (-1)^\Tk \frac{x^{2\Tk}}{2\Tk!} \To$
    \item $\Ds \frac{1}{1-x} = 1 + x + x^2 + x^3 + \operatorname{o}(x^3)
      = \sum_{\Tk=0}^{\Tn} x^\Tk \To$
    \item $\ds \arcsin x = x + \frac{x^3}{6} + \frac{3x^5}{40} + \operatorname{o}(x^6)
      =\sum_{\Tk=0}^{\Tn} \frac{(2\Tk)!}{4^\Tk (\Tk!)^2 (2\Tk+1)} x^{2\Tk+1} \To$
    % \item $\ds \sqrt{1+x} = 1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16} - \frac{5x^4}{128} + \cdots$
  \end{enumerate}
\end{document}

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