One option is to use \newcommand
. Add the following lines to the preamble of your document
\newcommand{\pd}[2]{\frac{\partial{#1}}{\partial{#2}}}
\newcommand{\pdd}[2]{\frac{\partial^2{#1}}{\partial{#2}^2}}
These can be used within the document as follows
\pd u t = \pdd u x
Result:

Of course, \pd{u}{x}
is preferable aesthetically and often the only correct syntax, but I am mainly trying to illustrate a minimal method to print the derivatives.
EDIT: A command for mixed derivatives and a generalization of the double derivative command to include multiple derivatives
\newcommand{\pdd}[3]{\frac{\partial^#1{#2}}{\partial{#3}^#1}}
\newcommand{\pdm}[6]{\frac{\partial^#1{#2}}{\partial{#4}^#3 \partial{#6}^#5}}
Then
\pd y t = \pdd 4 y t + \pdm 5y 2x 3t
gives

physics
for other i temporary don't remember a name :-(\partial
, for example.