\newcommand{\deriv}[2][]{\dfrac{\partial #1}{\partial #2}}
\newcommand{\tderiv}[2][]{\tfrac{\partial #1}{\partial #2}}
\newcommand{\fderiv}[2][]{\dfrac{d #1}{d #2}}
\newcommand{\derivII}[3][]{\dfrac{\partial^2 #1}{\partial #2 \partial #3}}
\newcommand{\tderivII}[3][]{\tfrac{\partial^2 #1}{\partial #2 \partial #3}}
\newcommand{\derivIII}[4][]{\dfrac{\partial^3 #1}{\partial #2 \partial #3 \partial #4}}
\newcommand{\tderivIII}[4][]{\tfrac{\partial^3 #1}{\partial #2 \partial #3 \partial #4}}
\newcommand{\derivIV}[5][]{\dfrac{\partial^4 #1}{\partial #2 \partial #3 \partial #4 \partial #5}}

\newcommand{\half}{\tfrac{1}{2}}  % Macro for 1/2

\newcommand{\ud}{\mathrm{d}}

\newcommand{\op}{\circ}
\newcommand{\dq}{\dot{q}}
\newcommand{\ddq}{\ddot{q}}

\newcommand{\dt}{\Delta t}

