After a backslash did its work

Never use braces just to terminate a command in math mode, they form an empty node, rather like \mbox{} and so affect the space. In your example just use

$\rho+\Delta A$

An example with a clear spacing difference would be

enter image description here



$\rho \leq{} + 2 A$

$\rho \leq + 2 A$

In general all letters qualify to follow a backslash and make up a command name. These letters are a-z and A-Z by default. Some commands like \makeatletter or \ExplSyntaxOn add other characters as letters (e.g. @ or _ and :). If one of those characters follows the command name you have to put some boundary there. The easiest way to do this is to follow your command by a space (as mentioned by @DavidCarlisle).

Examples (if you want more details consider reading this post):

\DeltaA % this would be one command, use
\Delta A % instead as this has the "boundary", but
\Delta1 % would not be a problem as 1 is no "letter" by default
\Delta@ % see above, but
\makeatletter\Delta@\makeatother % would call for a macro \Delta@

If math mode is concerned you are done using the space. In text mode you may want to consider that the space after a command is absorbed. So in text mode you may want to use \textDelta{} (or if you want to ensure the space \textDelta\).