When would you apply the product rule in differentiation and how do you do this?

The product rule is used to differentiate a function when it is in the form y= u(x)v(x). To use the rule you differentiate u(x) and multiply that by v(x), and then add that to the differential of v(x) multiplied by u(x). This gives you the differential of y in the form dy/dx= vdu/dx + u*dv/dx.