The Product Rule

Say you have a function like this one:

                                               

The product rule simply states that if the function can be split up into a sub-function multiplied by another sub-function, then:

                                                 

Here u and v are the sub-functions, and  and  are their derivatives with respect to x.

Remember:

                                                           

Here’s an example of using the product rule:

Product rule question

Find the derivative of  with respect to x.

Solution

Let u = 3x2 + 2x                                                        Let v = 2x4 – 5. Then:                                                                                                                       So:                                      This you can then expand out and simplify to give:

Be careful when you write ‘u’ and ‘v’ – it is easy to mix them up!  You could use the letters ‘a’ and ‘b’ just as easily, ‘u’ and ‘v’ are used here because many textbooks use them.