\documentclass[preview,border=1pt]{standalone}
\usepackage{amsmath,amssymb,amsthm}


\begin{document}

\newcommand{\diff}[1]{\,d#1}
\newcommand{\diffr}[1]{\,\mathrm{d}#1}
Inline: $\iint xy \diff x \diff y = \iint xy \diffr x \diffr y$. 
Display: \[ \iint xy \diff x \diff y = \iint xy \diffr x \diffr y. \]
\end{document}