LaTeX equation

The fraction form using d:

\[
y = \begin{cases}
3 & (x = 1) \\
4 & (x < 4)
\end{cases}
\]

\[ y = \begin{cases} 3 & (x = 1) \\ 4 & (x < 4) \end{cases} \]

The last two are the partial differentiation.

Apostrophe

\[
f' \\~\\
f'' \\~\\
f'(x) \\~\\
f^\prime \\~\\
f^{\prime} \\~\\
f^{\prime}(x)
\]

\[ f' \\~\\ f'' \\~\\ f'(x) \\~\\ f^\prime \\~\\ f^{\prime} \\~\\ f^{\prime}(x) \]

Dot

The vector differentiation often use dots particularly in physics. One dot means the first derivative and two dots means the second derivative.

\[
\dot{a} \\~\\
\ddot{a}
\]

\[ \dot{a} \\~\\ \ddot{a} \]

LaTeX equation

LaTeX