if I were you I'd rather stick with writing real Perl code with $x variables

This regex approach is quite dangerous, since any x will be replaced, for instance exp(x) could become e$xp($x)

so at least try to replace at word boundaries \b with $str =~ s/\bx\b/\$x/g

demonstration:

  DB<134> "ex p"=~s/\bx\b/\$x/r
 => "ex p"

  DB<135> "e x p"=~s/\bx\b/\$x/r
 => "e \$x p"
[download]

There are some ways to tweak Perl's parser though, but they are far too advanced for your level now.

i.e. using a function sub a {$a} to replace x with a (x is already an operator) and maybe catch and translate stuff like sin2 in AUTOLOAD.