can anybody tell me how the Leibniz's rule for the differentiation work?

Question

I have not studied partial differentiation yet...

Answer 1

This is better known as the product rule.
http://en.wikipedia.org/wiki/Product_rule

Take f(x)=x^2 sin x for instance.

f'=sin x d/dx x^2 + x^2 d/dx sinx

Answer 2

There are several different rules for differentiation attributed to Leibniz. One os sexplained at
http://www.karlscalculus.org/calc8_4.html
Another has to do with differentiation of a definite integral, with respect to a variable that appears in the integrand and/or the limits of integeration.