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\title{Table of derivatives}
\author{Your name}
\begin{document}
\maketitle
The \emph{derivative} of a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined as
\[
\frac{df}{dx}(x_0) = \lim_{h \rightarrow 0} \frac{f(x_0 + h) - f(x_0)}{h}.
\]

\begin{gather*}
\begin{array}{|c|}
\hline
\textbf{The derivative of linear combinations. }\\ \hline \\ \text{If }f,g \text{ are functions and }c \in \mathbb{R} \text{ then} \\
\displaystyle \frac{df + c\cdot g}{dx}= \frac{df}{dx}  + c \cdot \frac{dg}{dx} \\ \\
\hline
\textbf{The derivative of a product. }\\ \hline \\ \text{If }f,g \text{ are functions} \\
\displaystyle \frac{df\cdot g }{dx}= g \cdot \frac{df}{dx} + f \cdot \frac{dg}{dx} \\ \\
\hline
\textbf{The chain rule. }\\ \hline \\\text{If }f,g \text{ are functions} \\
\displaystyle \frac{dg(f)}{dx} = \frac{dg}{df} \cdot \frac{df}{dx}\\ \\
\hline
\textbf{Derivatives of common functions.} \\
\hline
{ \renewcommand{\arraystretch}{2}\renewcommand{\tabcolsep}{0.3cm}
\begin{array}{| c | c | c |}
\displaystyle \frac{d x^k}{dx} = k x^{k-1}, k \in \mathbb{R} &\displaystyle \frac{d a^x}{dx} = a^x\ln_a(x) & \displaystyle \frac{d \ln_a(x)}{dx} = \frac{1}{\ln(a) \cdot x}\\
\hline
\displaystyle \frac{d \sin(x)}{dx} = cos(x) & \displaystyle \frac{d \cos(x)}{dx} = -sin(x) & \displaystyle \frac{d \tan}{dx} = sec(x)^2  \\
\hline
\displaystyle \frac{d \cot(x)}{dx} = -csc(x)^2 & \displaystyle \frac{d \sec(x)}{dx} = \sec(x)\cdot \tan(x) & \displaystyle \frac{d \csc}{dx} = -\csc(x)\cdot \cot(x)  \\
\hline
\displaystyle \frac{d \arcsin(x)}{dx} = \frac{1}{\sqrt{1 - x^2}}& \displaystyle \frac{d \arccos(x)}{dx} = \frac{-1}{\sqrt{1 - x^2}} & \displaystyle \frac{d \arctan}{dx} = \frac{1}{1 + x^2}\\
\hline
\displaystyle \frac{d arccot(x)}{dx} = \frac{-1}{1 + x^2} & \displaystyle \frac{d arcsec(x)}{dx} = \frac{1}{|x|\sqrt{1 - x^2}} & \displaystyle \frac{d arccsc}{dx} = \frac{-1}{|x|\sqrt{1 - x^2}}  \\
\end{array}}\\
\hline
\end{array}
\end{gather*}
Recall that the absolute value function is defined in the following way
$$
|x| = \begin{cases}
x & \text{if } x>0 \\
-x & \text{otherwise}
\end{cases}
$$


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