From a2b52243d89a21acb35c3c50dad793aa8113e162 Mon Sep 17 00:00:00 2001
From: Didnelpsun <48906416+Didnelpsun@users.noreply.github.com>
Date: Sun, 24 Jan 2021 23:02:30 +0800
Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0?=
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---
 advanced-math/1-perpare/perpare.tex           |  8 +--
 .../function-and-limit.synctex(busy)          |  0
 ...ariable-function-differential-calculus.tex | 55 +++++++++++++++++--
 3 files changed, 53 insertions(+), 10 deletions(-)
 create mode 100644 advanced-math/3-function-and-limit/function-and-limit.synctex(busy)

diff --git a/advanced-math/1-perpare/perpare.tex b/advanced-math/1-perpare/perpare.tex
index a3fc712..900073a 100644
--- a/advanced-math/1-perpare/perpare.tex
+++ b/advanced-math/1-perpare/perpare.tex
@@ -531,7 +531,7 @@ $$
 \begin{tikzpicture}[scale=0.9]
     \draw[-latex](-3,0) -- (3,0) node[below]{$x$};
     \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$};
-    \draw[black, thick, domain=-3:3] plot (\x,{pi/2-rad(atan(\x))}) node[right]{$\rm{arccot}(x)$};
+    \draw[black, thick, domain=-3:3] plot (\x,{pi/2-rad(atan(\x))}) node[right]{$\rm{arccot}(\textit{x})$};
     \filldraw[black] (0,0) node[below]{$O$};
     \draw[black, densely dashed](-3,pi) -- (3,pi);
     \filldraw[black] (-0.5,pi/2-0.5) node{$\dfrac{\pi}{2}$};
@@ -541,11 +541,11 @@ $$
 
 \begin{enumerate}
     \item 特殊函数值：$\arctan 0=0$，$\arctan\dfrac{\pi}{6}=\dfrac{\sqrt{3}}{3}=$，$\arctan 1=\dfrac{\pi}{4}$，$\arctan\sqrt{3}=\dfrac{\pi}{3}$，$\rm{arccot}0=\dfrac{\pi}{2}$，$\rm{arccot}\sqrt{3}=\dfrac{\pi}{6}$，$\rm{arccot}1=\dfrac{\pi}{4}$，$\rm{arccot}\dfrac{\sqrt{3}}{3}=\dfrac{\pi}{3}$。
-    \item 定义域：$(-\infty, +\infty)$，值域：$\arctan x:[-\dfrac{\pi}{2},+\dfrac{\pi}{2}]$，$\rm{arccot}x:[0,\pi]$。
+    \item 定义域：$(-\infty, +\infty)$，值域：$\arctan x:[-\dfrac{\pi}{2},+\dfrac{\pi}{2}]$，$\rm{arccot}\,\textit{x}:[0,\pi]$。
     \item 单调性：$y=\arctan x$单调增，$y=\rm{arccot}x$单调减。
     \item 奇偶性：$y=\arctan x$为奇函数。
-    \item 有界性：$\vert\arctan x\vert\leqslant\dfrac{\pi}{2}$，$0\leqslant\rm{arccot}x\leqslant\pi$。
-    \item 性质：$\arctan x+\rm{arccot}x=\dfrac{\pi}{2}(-\infty<x<+\infty)$
+    \item 有界性：$\vert\arctan x\vert\leqslant\dfrac{\pi}{2}$，$0\leqslant\rm{arccot}\,\textit{x}\leqslant\pi$。
+    \item 性质：$\arctan x+\rm{arccot}\,\textit{x}=\dfrac{\pi}{2}(-\infty<x<+\infty)$
 \end{enumerate}
 
 \subparagraph{初等函数} \leavevmode \bigskip
diff --git a/advanced-math/3-function-and-limit/function-and-limit.synctex(busy) b/advanced-math/3-function-and-limit/function-and-limit.synctex(busy)
new file mode 100644
index 0000000..e69de29
diff --git a/advanced-math/4-single-variable-function-differential-calculus/single-variable-function-differential-calculus.tex b/advanced-math/4-single-variable-function-differential-calculus/single-variable-function-differential-calculus.tex
index 5c09143..9d538e8 100644
--- a/advanced-math/4-single-variable-function-differential-calculus/single-variable-function-differential-calculus.tex
+++ b/advanced-math/4-single-variable-function-differential-calculus/single-variable-function-differential-calculus.tex
@@ -533,14 +533,57 @@ $\therefore \dfrac{y^{(6)}(0)}{6!}=-\dfrac{1}{6}\Rightarrow y^{(6)}(0)=-5!=-120$
 
 \subsection{基本求导公式}
 
+\subsubsection{对幂指函数}
+
 \begin{center}
-    \begin{tabular}{|c|c|}
+    \begin{tabular}{|c|c|c|c|}
         \hline
-        原函数 & 导函数 \\ \hline
-        $C$ & $0$ \\ \hline
-        $n^x$ & $n^x\ln n$ \\ \hline
-        $\log_ax$ & $\dfrac{1}{x\ln a}$ \\ \hline
-        $\ln x(\ln\vert x\vert)$ & $\dfrac{1}{x}$ \\
+        原函数 & 导函数 & 原函数 & 导函数\\ \hline
+        $C$ & $0$ & $n^x$ & $n^x\ln n$ \\ \hline
+        $\log_ax$ & $\dfrac{1}{x\ln a}$ & $\ln x=\ln\vert x\vert$ & $\dfrac{1}{x}$ \\ \hline
+        $x^n$ & $nx^{n-1}$ & $\sqrt[n]{x}$ & $\dfrac{x^{-\frac{n-1}{n}}}{n}$ \\ \hline
+        $\dfrac{1}{x^n}$ & $-\dfrac{n}{x^{n+1}}$ & & \\ 
+        \hline
+    \end{tabular}
+\end{center}
+
+\subsubsection{三角与反三角函数}
+
+\begin{center}
+    \begin{tabular}{|c|c|c|c|}
+        \hline
+        原函数 & 导函数 & 原函数 & 导函数\\ \hline
+        $\sin x$ & $\cos x$ & $\cos x$ & $-\sin x$ \\ \hline
+        $\tan x$ & $\dfrac{1}{\cos^2x}=\sec^2x$ & $\cot x$ & $\dfrac{1}{\sin^2x}=\csc^2x$ \\ \hline
+        $\sec x$ & $\sec x\tan x$ & $\csc x$ & $-\csc x\cot x$ \\ \hline
+        $\arcsin x$ & $\dfrac{1}{1-x^2}$ & $\arccos x$ & $-\dfrac{1}{1-x^2}$ \\ \hline
+        $\arctan x$ & $\dfrac{1}{1+x^2}$ & $\rm{arccot}\,\textit{x}$ & $-\dfrac{1}{1+x^2}$ \\ \hline
+        $\rm{arcsec}\,\textit{x}$ & $\dfrac{1}{x\sqrt{x^2-1}}$ & $\rm{arccsc}\,\textit{x}$ & $-\dfrac{1}{x\sqrt{x^2-1}}$ \\ \hline
+        \hline
+    \end{tabular}
+\end{center}
+
+\subsubsection{双曲与反双曲函数}
+
+\begin{itemize}
+    \item 双曲正弦：$\rm{sinh}\,\textit{x}=\rm{sh}\,\textit{x}=\dfrac{e^x-e^{-x}}{2}$。
+    \item 双曲余弦：$\rm{cosh}\,\textit{x}=\rm{ch}\,\textit{x}=\dfrac{e^x+e^{-x}}{2}$。
+    \item 双曲正切：$\rm{tanh}\,\textit{x}=\rm{th}\,\textit{x}=\dfrac{\rm{sinh}\,\textit{x}}{\rm{cosh}\,\textit{x}}=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}$。
+    \item 双曲余切：$\rm{coth}\,\textit{x}=\dfrac{\rm{cosh}\,\textit{x}}{\rm{sinh}\,\textit{x}}=\dfrac{e^x+e^{-x}}{e^x-e^{-x}}$。
+    \item 双曲正割：$\rm{sech}\,\textit{x}=\dfrac{1}{\rm{cosh}\,\textit{x}}=\dfrac{2}{e^x+e^{-x}}$。
+    \item 双曲余割：$\rm{csch}\,\textit{x}=\dfrac{1}{\rm{sinh}\,\textit{x}}=\dfrac{2}{e^x-e^{-x}}$。
+    \item 反双曲正弦：$\rm{arcsinh}\,\textit{x}=\ln\left(x+\sqrt{x^2+1}\right)$。
+    \item 反双曲余弦：$\rm{arccosh}\,\textit{x}=\ln\left(x+\sqrt{x^2-1}\right)$。
+    \item 反双曲正切：$\rm{arctanh}\,\textit{x}=\dfrac{1}{2}\ln\left(\dfrac{1+x}{1-x}\right)$。
+\end{itemize}
+
+\begin{center}
+    \begin{tabular}{|c|c|c|c|}
+        \hline
+        原函数 & 导函数 & 原函数 & 导函数\\ \hline
+        $\rm{sinh}\,\textit{x}$ & $\rm{cosh}\,\textit{x}$ & $\rm{cosh}\,\textit{x}$ & $\rm{sinh}\,\textit{x}$ \\ \hline
+        $\rm{tanh}\,\textit{x}$ & $\dfrac{1}{\rm{cosh}\,\textit{x}^2}$ & $\rm{arcsinh}\,\textit{x}$ & $\dfrac{1}{\sqrt{x^2+1}}$ \\ \hline
+        $\rm{arccosh}\,\textit{x}$ & $\dfrac{1}{\sqrt{x^2-1}}$ & $\rm{arctan}\,\textit{x}$ & $\dfrac{1}{1-x^2}$ \\
         \hline
     \end{tabular}
 \end{center}