The product rule is used to differentiate the product of two functions.

Given y = f(x).g(x), then

                                    

 

Differentiate, with respect to x,

       (i)    (x - 1)(x + 2)

       (ii)   (2x + 1)(x² + 4x + 7)

       (iii)  (x² - 1)(x - 2)⁴

       (iv)  (x - 2)^1/2(x³ + 1)

 

Solutions:

 

 (i)   d/dx [(x - 1)(x + 2)]

            = (x - 1) + (x + 2)

            = 2x - 1

       Alternatively, expand and use the sum & difference rules.

 

 (ii)  d/dx [(2x + 1)(x² + 4x + 7)]

            = (2x + 4)(2x + 1) + 2(x² + 4x + 7)

            = 4x² + 10x + 4 + 2x² + 8x + 14

            = 2(x² + 9x + 7)

 

 (iii) d/dx [(x² - 1)(x - 2)⁴]

            = 2x(x - 2)⁴ + 4(x - 2)³(x² - 1)

            = 2(x - 2)³[x(x - 2) + 2(x² - 1)]

            = 2(x - 2)³(3x² - 2x - 1)

 

 (iv)

             

 

 

Product Rule Extended

The product rule as stated above is used to differentiate a function that is a product of two functions. This rule can be extended to products of more than 2 functions. Here's an example for the product of 3 functions.

 

If y = f(x).g(x).h(x), then

          

 

Find the derivative of x(x + 1)(x² + 2)

 

d/dx [x(x + 1)(x² + 2)]

       = (x + 1)(x² + 2) + x(x² + 2) + 2x²(x + 1)

       = x³ + x² + 2x + 2 + x³ + 2x + 2x³ + 2x²

       = 4x³ + 3x² + 4x + 2