The Product Rule is used when differentiating two functions that are being multiplied together. It can be used by multiplying each function by the derivative of the other and adding.  

If y=u*v then

dy/dx= u*dv/dx + v*du/dx  

To illustrate this rule look at the example below: 

y=x²*e^3x

u=x²  v=e^3x      du/dx= 2x    dv/dx= 3e^3x

Therefore dy/dx= (x²)*(3e^3x)+ (e^3x)*(2x)  

               dy/dx= 3x²e^3x + 2xe^3x