partial differentiation

The partial differentiation of an equation with respect to $x$ ($\frac{\partial }{\partial x}$) can be obtained by differentiating the equation with respect to $x$ and treating other variables as if they were constants. For example, $\frac{\partial }{\partial x}(3x+2xy)=3+2y$ .

For partial differentiation involving underlying variables, use chain rule and chain rule dependency diagram (e.g. $\frac{\partial y}{\partial t}$ when $y$ is defined not in terms of $t$, but some other variables that can be defined with $t$)