对齐环境后无法摆脱 vspace

Question 1

不要使用\center，而是\centering，并使显示前后的空格为零。由于您使用align，\noindent因此是必要的。不要在行后留下空行\cheattitle。

另外，不要留下尾随\\，align这会在显示中产生一个空白行。

我建议amsart直接支持8pt。

\documentclass[8pt]{amsart}
\usepackage[T1,T2A]{fontenc}
\usepackage[english, ukrainian]{babel}

\usepackage{geometry}
\geometry{
  paper=a5paper,
  inner=0cm, outer=0cm, top=0cm, bottom=0cm, 
  bindingoffset=0cm, 
  includehead=false,
  includefoot=false,
  landscape,
  twoside
}

\usepackage{amsmath}
\usepackage{booktabs}

\newcommand{\cheattitle}[1]{%
  \par{\sffamily\slshape\centering#1\par}\noindent\ignorespaces
}


\begin{document}
\setlength{\abovedisplayskip}{0pt}
\setlength{\abovedisplayshortskip}{0pt}
\setlength{\belowdisplayskip}{0pt}
\setlength{\belowdisplayshortskip}{0pt}

\cheattitle{Основні правила диференціювання}
\begin{align*}
    &(u+v)'=u'+v' && (u-v)'=u'-v' & & (uv)'=u'v+uv' &  & (Cu)'=Cu'\\
    & (uvw)'=u'vw+uv'w+uvw' & & \left(\frac{u}{v}\right)' =\frac{u'v-uv'}{v^2},\ v\neq 0 & & \left(\frac{C}{v}\right)'=-C\frac{v'}{v^2},\ v\neq 0 & & \left(\frac{u}{C}\right)'=\frac{1}{C}u'
\end{align*}

\cheattitle{Таблиця похідних}
\begin{align*}
    & C'=0 & & (u^n)'=nu^{n-1}u' & & \left(\sqrt{u}\right)=\frac{1}{2\sqrt{u}}u' & & \left(\frac{1}{u}\right)'=-\frac{1}{u^2}u' & & \left(a^u\right)'=a^u\ln a\cdot u' \\
    & (e^u)'=e^uu' & & \left(\log_au\right)'=\frac{1}{u\ln a} u' & & \left(\ln u\right)'=\frac{1}{u}u' & & \left(\sin u\right)'=\cos u\cdot u' & & \left(\cos u\right)'=-\sin u\cdot u'\\
    & \left(\tg u\right)' = \frac{1}{\cos^2u}u' & &\left(\ctg u\right)' = \frac{1}{\sin^2u}u' & & \left(\arcsin u\right)' = \frac{1}{\sqrt{1-u^2}}u' & & \left(\arccos u\right)'= -\frac{1}{\sqrt{1-u^2}} & & \left(\arcctg u\right)' = \frac{1}{i+u^2}u'\\
    & \left(\arcctg u\right)'=-\frac{1}{1+u^2}u' & & \left(\sh u\right)' = \ch u \cdot u' & & \left(\ch u\right)' = \sh u\cdot u' & & \left(\th u\right)' = \frac{1}{\ch^2u}u' & & \left(\cth u\right)' = -\frac{1}{\sh^2u}u'
\end{align*}

\cheattitle{Формула Тейлора}
\begin{align*}
    f(x) = f(0)+\frac{f'(0)}{1!}x+\frac{f''(0)}{2!}x^2+\dots+\frac{f^{n}(0)}{n!}x^n+\frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}
\end{align*}
\end{document}

或者，article使用fontsize。

\documentclass{article}
\usepackage[T1,T2A]{fontenc}
\usepackage[english, ukrainian]{babel}
\usepackage[fontsize=8pt]{fontsize}

\usepackage{geometry}
\geometry{
  paper=a5paper,
  inner=0cm, outer=0cm, top=0cm, bottom=0cm, 
  bindingoffset=0cm, 
  includehead=false,
  includefoot=false,
  landscape,
  twoside
}

\usepackage{amsmath}
\usepackage{booktabs}

\newcommand{\cheattitle}[1]{%
  \par{\sffamily\slshape\centering#1\par}\noindent\ignorespaces
}


\begin{document}
\setlength{\abovedisplayskip}{0pt}
\setlength{\abovedisplayshortskip}{0pt}
\setlength{\belowdisplayskip}{0pt}
\setlength{\belowdisplayshortskip}{0pt}

\cheattitle{Основні правила диференціювання}
\begin{align*}
    &(u+v)'=u'+v' && (u-v)'=u'-v' & & (uv)'=u'v+uv' &  & (Cu)'=Cu'\\
    & (uvw)'=u'vw+uv'w+uvw' & & \left(\frac{u}{v}\right)' =\frac{u'v-uv'}{v^2},\ v\neq 0 & & \left(\frac{C}{v}\right)'=-C\frac{v'}{v^2},\ v\neq 0 & & \left(\frac{u}{C}\right)'=\frac{1}{C}u'
\end{align*}

\cheattitle{Таблиця похідних}
\begin{align*}
    & C'=0 & & (u^n)'=nu^{n-1}u' & & \left(\sqrt{u}\right)=\frac{1}{2\sqrt{u}}u' & & \left(\frac{1}{u}\right)'=-\frac{1}{u^2}u' & & \left(a^u\right)'=a^u\ln a\cdot u' \\
    & (e^u)'=e^uu' & & \left(\log_au\right)'=\frac{1}{u\ln a} u' & & \left(\ln u\right)'=\frac{1}{u}u' & & \left(\sin u\right)'=\cos u\cdot u' & & \left(\cos u\right)'=-\sin u\cdot u'\\
    & \left(\tg u\right)' = \frac{1}{\cos^2u}u' & &\left(\ctg u\right)' = \frac{1}{\sin^2u}u' & & \left(\arcsin u\right)' = \frac{1}{\sqrt{1-u^2}}u' & & \left(\arccos u\right)'= -\frac{1}{\sqrt{1-u^2}} & & \left(\arcctg u\right)' = \frac{1}{i+u^2}u'\\
    & \left(\arcctg u\right)'=-\frac{1}{1+u^2}u' & & \left(\sh u\right)' = \ch u \cdot u' & & \left(\ch u\right)' = \sh u\cdot u' & & \left(\th u\right)' = \frac{1}{\ch^2u}u' & & \left(\cth u\right)' = -\frac{1}{\sh^2u}u'
\end{align*}

\cheattitle{Формула Тейлора}
\begin{align*}
    f(x) = f(0)+\frac{f'(0)}{1!}x+\frac{f''(0)}{2!}x^2+\dots+\frac{f^{n}(0)}{n!}x^n+\frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}
\end{align*}
\end{document}

不要添加textwidth=\textwidth和textheight=\textheight：它们只会导致烦人的警告。

还请注意，这\usepackage[utf8]{inputenc}不再是必要的，您应该加载（最后）西里尔文的 T2A 编码。

最后，\sf已经被弃用大约 30 年了。

Answer