I'm studying calculus and sometimes I find it strange to treat dx(differential) like numbers! Substitution rule would be a good example. ( I will use the first example in this website http://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleIndefinite.aspx ) u = 6x^3 + 5 becomes du = 18*x^2*dx.
But I am curious whether there are mathematical grounds for treating differentials like numbers! If there are, can you tell me which field of mathematics deals with such kind of things?
The grounds for treating infinitesimal $dx$ as a number is the rigorous theory of infinitesimals first developed by Abraham Robinson in the 1960s, based on earlier work by Thoralf Skolem, Edwin Hewitt, and Jerzy Łoś.
The OP asked for sources where one can learn about this approach. The best source is the textbook Elementary Calculus by Keisler where this is explained at a level accessible to a freshman.
Leibniz used infinitesimal numbers extensively specifically in the form of differentials which he denoted $dx$, $dy$. As it turned out his procedures and his theoretical justification for manipulating these were more soundly based than Bishop Berkeley's criticism thereof; see this site for abundant literature on the subject. In my teaching experience this approach is more successful with the students.