product is the result of multiplying two terms or expressions together. e.g. (x + 2)(x − 3) is a product.

To differentiate a product, brackets can be expanded and the function differentiated term by term.

However, this can be a long process and a quicker way to differentiate a product may be to use the Product Rule.

Product Rule Formula

This states:

(f . g )'   =   f '. g  +  f . g'


The Product Rule looks complex, but with practice, which is essential, it should become more straightforward.

Example To differentiate h(x) = 3xsin x

Let f(x) = 3x2 and g(x) = sin x

Therefore, f '(x) = 6x and g '(x) = cos x

h '(x) = f '. g + f . g '

=6x . sin x + 3x2 . cos x 

= 6x sin x + 3x2 cos x

Alternative Product Rule Formula

The Product Rule can be written in Y12_The_Product_Rule_01.gif notation:

If y = u .v


Example To differentiate y = 3x2 sin x

Let u = 3x2 and v = sin x

Therefore, Y12_The_Product_Rule_03.gif = 6x and Y12_The_Product_Rule_04.gif = cos x


Y12_The_Product_Rule_06.gif= 3x2 . cos x + sin x . 6x

     = 3x2 cos x + 6x sin x

(You should try to do the "Let u = ..." mentally. This becomes easier with practice and you should soon be able to write out the answer in one line!)