A list of common formulas, not in any way complete.
\[
\begin{align}
(c)' &= 0 &
(x)' &= 1 \\
(x^n)' &= n \cdot x^{n-1} &
\left(\sqrt{x}\right)' &= \frac{1}{2\sqrt{x}} \\
(e^x)' &= e^x &
(a^x)' &= a^x \cdot \ln a \\
(\ln x)' &= \frac{1}{x} &
(\log_a x)' &= \frac{1}{x \cdot \ln a}
\end{align}
\]
\[
\begin{align}
(f + g)'(x) &= f'(x) + g'(x) &
(c \cdot f)'(x) &= c \cdot f'(x)
\end{align}
\]
\[
\begin{align}
(f \cdot g)'(x) &= f'(x) \cdot g(x) + f(x) \cdot g'(x) \\
(f_1 \cdot f_2 \cdot \ldots \cdot f_n)'(x) &= f'_1(x) \cdot f_2(x) \cdot \ldots \cdot f_n(x) \\
&+ f_1(x) \cdot f'_2(x) \cdot \ldots \cdot f_n(x) \\
&\dots \\
&+ f_1(x) \cdot f_2(x) \cdot \ldots \cdot f'_n(x)
\end{align}
\]
\[
\begin{align}
\left( \frac{1}{f} \right)'(x) &= \frac{-f'(x)}{f(x)^2} \\
\left( \frac{f}{g} \right)'(x) &= \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{g(x)^2} \\
\end{align}
\]
\[
\begin{equation}
(g \circ f)'(x) = g'(f(x)) \cdot f'(x) \tag{chain rule}
\end{equation}
\]
\[
\begin{align}
(\sin x)' &= \cos x &
(\cos x)' &= -\sin x \\
(\tan x)' &= \frac{1}{\cos^2 x} &
(\cot x)' &= \frac{-1}{\sin^2 x} \\
(\arcsin x)' &= \frac{1}{\sqrt{1 - x^2}} &
(\arccos x)' &= \frac{-1}{\sqrt{1 - x^2}} \\
(\arctan x)' &= \frac{1}{1 + x^2} &
(\operatorname{arccot} x)' &= \frac{-1}{1 + x^2}
\end{align}
\]