\newcommand{\pderiv}[1] {\frac{\partial}{\partial #1}}
\newcommand{\pderivf}[2] {\frac{\partial #1}{\partial #2}}
\newcommand{\pderivs}[1] {\frac{\partial^2}{\partial {#1}^2}}
\newcommand{\pderivfs}[2] {\frac{\partial {#1}^2}{\partial {#2}^2}}
\newcommand{\pderivho}[2] {\frac{\partial^2}{\partial #1 \partial {#2}}}
\newcommand{\pderivfho}[3] {\frac{\partial^2 #1}{\partial #2 \partial #3}}
\newcommand{\deriv}[1] {\frac{\mathrm{d}}{\mathrm{d} #1}}
\newcommand{\derivs}[1] {\frac{\mathrm{d}^2}{\mathrm{d} {#1}^2}}
\newcommand{\derivf}[2] {\frac{\mathrm{d} #1}{\mathrm{d} #2}}
\newcommand{\derivfs}[2] {\frac{\mathrm{d} {#1}^2}{\mathrm{d} {#2}^2}}
\newcommand{\paren}[1] {\left( #1 \right)}
\newcommand{\anglebrack}[1] {\left\langle #1 \right\rangle}
\newcommand{\abrack}[1] {\left\langle #1 \right\rangle}
\newcommand{\curly}[1] {\left\{ #1 \right\}}
\newcommand{\curlybrack}[1] {\left\{ #1 \right\}}
\newcommand{\cbrack}[1] {\left\{ #1 \right\}}
\newcommand{\squarebrack}[1] {\left[ #1 \right]}
\newcommand{\sqbrack}[1] {\left[ #1 \right]}
\newcommand{\sbrack}[1] {\left[ #1 \right]}
\newcommand{\brackets}[1] {\left[ #1 \right]}
\newcommand{\d}[1]{\mathrm{d}#1}
\newcommand{\bb}[1] {\mathbb{#1}}