We sometimes come across quantities that are functions of more than one independent variable. Thus, we may find a case where ${\displaystyle y}$ depends on two other variable quantities, one of which we will call ${\displaystyle u}$ and the other ${\displaystyle v}$. In symbols

Take the simplest concrete case.

Let

What are we to do? If we were to treat ${\displaystyle v}$ as a constant, and differentiate with respect to ${\displaystyle u}$, we should get

or if we treat ${\displaystyle u}$ as a constant, and differentiate with respect to ${\displaystyle v}$, we should have:

The little letters here put as subscripts are to show which quantity has been taken as constant in the operation.

Another way of indicating that the differentiation has been performed only partially, that is, has been performed only with respect to one of the independent