LaTeX calculus

The fraction form using d:

\[
\frac{ d }{ dx } f \\~\\
\frac{ df }{ dx } \\~\\
\frac{\partial y}{\partial x} \\~\\
\frac{\partial{y}}{\partial{x}}
\]

\[ \frac{ d }{ dx } f \\~\\ \frac{ df }{ dx } \\~\\ \frac{\partial y}{\partial x} \\~\\ \frac{\partial{y}}{\partial{x}} \]

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.

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

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