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[PS] Add tables
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\rmvspace
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\subsection{Verschiedene Funktionen}
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\subsubsection{Logarithmen}
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\begin{itemize}
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\item \textit{(Basiswechsel)} $\log_a(x) = \frac{\ln(x)}{\ln(a)}$
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\item \textit{(Potenzen)} $\log_a(x^y) = y\log_a(x)$
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\item \textit{(Div, Mul)} $\log_a(x \cdot (\div) y) = \log_a(x) +(-) \log_a(y)$
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\item $\log_a(1) = 0 \smallhspace \forall a \in \N$
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\end{itemize}
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\subsubsection{Trigonometrie}
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$\cot(\xi) = \displaystyle\frac{\cos(\xi)}{\sin(\xi)}, \tan(\xi) = \frac{\sin(\xi)}{\cos(\xi)}$
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$\sinh(x) := \frac{e^x - e^{-x}}{2} : \R \rightarrow \R$,
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$\cosh(x) := \frac{e^x + e^{-x}}{2} : \R \rightarrow [1, \infty]$,
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$\cosh(x) := \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} : \R \rightarrow [-1, 1]$
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\begin{enumerate}
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\item $\cos(x) = \cos(-x)$ \trand $\sin(-x) = -\sin(x)$
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\item $\cos(\pi - x) = -\cos(x)$ \trand $\sin(\pi - x) \sin(x)$
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\item $\sin(x + w) = \sin(x) \cos(w) + \cos(x) \sin(w)$
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\item $\cos(x + w) = \cos(x) \cos(w) - \sin(x) \sin(w)$
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\item $\cos(x)^2 + \sin(x)^2 = 1$
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\item $\sin(2x) = 2 \sin(x) \cos(x)$
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\item $\cos(2x) = \cos(x)^2 - \sin(x)^2$
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\end{enumerate}
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\shade{teal}{\tr{Values of trigonometric functions}{Werte der trigonometrischen Funktionen}}
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\begin{tables}{ccccc}{° & rad & $\sin(\xi)$ & $\cos(\xi)$ & $\tan(\xi)$}
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0° & $0$ & $0$ & $1$ & $1$ \\
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\hline
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30° & $\frac{\pi}{6}$ & $\frac{1}{2}$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{3}}{2}$ \\
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\hline
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45° & $\frac{\pi}{4}$ & $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{2}}{2}$ & $1$ \\
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\hline
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60° & $\frac{\pi}{3}$ & $\frac{\sqrt{3}}{3}$ & $\frac{1}{2}$ & $\sqrt{3}$ \\
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\hline
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90° & $\frac{\pi}{2}$ & $1$ & $0$ & $\varnothing$ \\
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\hline
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120° & $\frac{2\pi}{3}$ & $\frac{\sqrt{3}}{2}$ & $-\frac{1}{2}$ & $-\sqrt{3}$ \\
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\hline
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135° & $\frac{3\pi}{4}$ & $\frac{\sqrt{2}}{2}$ & $-\frac{\sqrt{2}}{2}$ & $-1$ \\
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\hline
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150° & $\frac{5\pi}{6}$ & $\frac{1}{2}$ & $-\frac{\sqrt{3}}{2}$ & $-\frac{\sqrt{3}}{2}$ \\
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\hline
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180° & $\pi$ & $0$ & $-1$ & $0$ \\
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\end{tables}
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\subsection{Tabelle von Auf- und Ableitungen}
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\begin{scriptsize}
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\begin{tables}{lll}{\tr{Antiderivative}{Stammfunktion} & \tr{Function}{Funktion} & \tr{Derivative}{Ableitung}}
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$\displaystyle \frac{x^{n + 1}}{n + 1}$ & $x^n$ & $n \cdot x^{n - 1}$ \\
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$\ln|x|$ & $\displaystyle \frac{1}{x} = x^{-1}$ & $\displaystyle -x^{-2} = -\frac{1}{x^2}$ \\[0.2cm]
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$\frac{2}{3} x^{\frac{3}{2}}$ & $\displaystyle \sqrt{x} = x^{\frac{1}{2}}$ & $\displaystyle \frac{1}{2 \cdot \sqrt{x}}$ \\[0.3cm]
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$\frac{n}{n + 1} x^{\frac{1}{n} + 1}$ & $\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}$ & $\frac{1}{n} x^{\frac{1}{n} - 1}$ \\[0.3cm]
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\hline \\[-0.2cm]
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$e^x$ & $e^x$ & $e^x$ \\
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$\exp(x)$ & $\exp(x)$ & $\exp(x)$ \\
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$\frac{1}{a \cdot (n + 1)}(ax + b)^{n + 1}$ & $(ax + b)^n$ & $n\cdot (ax + b)^{n - 1} \cdot a$ \\
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$x \cdot (\ln|x| - 1)$ & $\ln(x)$ & $\frac{1}{x} = x^{-1}$ \\
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$\displaystyle \frac{1}{\ln(a)}\cdot a^x$ & $a^x$ & $a^x \cdot \ln(a)$ \\
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$\frac{x}{\ln(a)} \cdot (\ln|x| - 1)$ & $\log_a|x|$ & $\displaystyle \frac{1}{x \cdot \ln(a)}$ \\[0.3cm]
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\hline \\[-0.2cm]
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$-\cos(x)$ & $\sin(x)$ & $\cos(x)$ \\
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$\sin(x)$ & $\cos(x)$ & $-\sin(x)$ \\
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$-\ln|\cos(x)|$ & $\tan(x)$ & $\displaystyle \frac{1}{\cos^2(x)}$ \\[0.3cm]
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$x \cdot \arcsin(x) + \sqrt{1 - x^2}$ & $\arcsin(x)$ & $\displaystyle\frac{1}{\sqrt{1 - x^2}}$ \\
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$x \cdot \arccos(x) - \sqrt{1 - x^2}$ & $\arccos(x)$ & $\displaystyle -\frac{1}{\sqrt{1 - x^2}}$ \\
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$\displaystyle x \cdot \arctan(x) - \frac{\ln(x^2 + 1)}{2}$ & $\arctan(x)$ & $\displaystyle \frac{1}{x^2 + 1}$ \\[0.2cm]
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$\ln|\sin(x)|$ & $\cot(x)$ & $\displaystyle -\frac{1}{\sin^2(x)}$ \\
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$\cosh(x)$ & $\sinh(x)$ & $\cosh(x)$ \\
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$\sinh(x)$ & $\cosh(x)$ & $\sinh(x)$ \\
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$\ln|\cosh(x)|$ & $\tanh(x)$ & $\displaystyle \frac{1}{\cosh^2(x)}$ \\
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& $\arcsinh(x)$ & $\frac{1}{\sqrt{1 + x^2}}$ \\
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& $\arccosh(x)$ & $\frac{1}{\sqrt{x^2 - 1}}$ \\
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& $\arctanh(x)$ & $\frac{1}{1 - x^2}$ \\
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\end{tables}
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\end{scriptsize}
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\shade{teal}{Weitere Ableitungen}
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\begin{tables}{cc}{$F(x)$ & $f(x)$}
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$\frac{1}{a} \ln|ax + b|$ & $\frac{1}{ax + b}$ \\
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$\frac{ax}{c} - \frac{ad - bc}{c^2} \ln|cx + d|$ & $\frac{a (cx + d) - c(ax + b)}{(cx + d)^2}$ \\
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$\frac{x}{2} f(x) + \frac{a^2}{2} \ln|x + f(x)|$ & $\sqrt{a^2 + x^2}$ \\
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$\frac{x}{2} f(x) - \frac{a^2}{2} \arcsin\left( \frac{x}{|a|} \right)$ & $\sqrt{a^2 - x^2}$ \\
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$\frac{x}{2} f(x) - \frac{a^2}{2} \ln|x + f(x)|$ & $\sqrt{x^2 - a^2}$ \\
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$\ln(x + \sqrt{x^2 \pm a^2})$ & $\frac{1}{\sqrt{x^2 \pm a^2}}$ \\
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$\arcsin \left( \frac{x}{|a|} \right)$ & $\frac{1}{\sqrt{x^2 - a^2}}$ \\
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$\frac{1}{a}\arctan \left( \frac{x}{|a|} \right)$ & $\frac{1}{a^2 - x^2}$ \\
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\end{tables}
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\begin{tables}{cc}{$F(x)$ & $f(x)$}
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$-\frac{1}{a} \cos(ax + b)$ & $\sin(ax + b)$ \\
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$\frac{1}{a} \sin(ax + b)$ & $\cos(ax + b)$ \\[1mm]
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\hline
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$x^x$ & $x^x \cdot (1 + \ln|x|)$ \\
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$(x^x)^x$ & $(x^x)^x \cdot (x + 2x\ln|x|)$ \\
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$x^{(x^x)}$ & $x^{(x^x)} \cdot (x^{x - 1} + \ln|x| \cdot x^x (1 + \ln|x|))$ \\
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\hline \\[-3mm]
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$\frac{1}{2}(x - \frac{1}{2} \sin(2x))$ & $\sin(x)^2$ \\[1mm]
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$\frac{1}{2}(x + \frac{1}{2} \sin(2x))$ & $\cos(x)^2$ \\
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\end{tables}
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\subsection{Limits}
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\includegraphics[width=1\columnwidth]{./assets/limits.png}
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