When and how do I use the product rule for differentiation?

As the name suggests, the product rule is used to differentiate a function in which a product of 2 expressions in x exists. This means the two expressions in x are multiplied by each other, even when the function is expressed in its simplest form. An example would be y=x³e^2x. The product rule is written by generalising one expression in x as u and the other as v: 

If y=u*v then

dy/dx= udv/dx + vdu/dx

This means that, to dfferentiate, we multiply each expression in x by the derivative of the other and add the results. This is illustrated by the example below: 

 y=x³e^2x

let u= x³                 v=e^2x

du/dx = 3x²            dv/dx= 2e^2x

for this example: 

                       dy/dx = u dv/dx + v du/dx

                                = x³*2e^2x +  e^2x*3x² 

                                = e^2x(2x³ + 3x²)