A 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
(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) = 3x2 sin 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 notation:
If y = u .v
Example To differentiate y = 3x2 sin x
Let u = 3x2 and v = sin x
Therefore, = 6x and = cos x
= 3x2 . cos x + sin x . 6x
= 3x2 cos x + 6x sin x