9.3 First Derivatives of the Product of Two Polynomials

9.3 Find the Derivatives of a Product using Product Rule
(A) The Product Rule

Method 1

If u(x) and v(x) are two functions of x and y = uv then

Example:


Method 2 (Differentiate Directly)



Example:
Given that y=(2x+3)(3 x 3 2 x 2 x), find  dy dx
Solution:
y=(2x+3)(3 x 3 2 x 2 x) dy dx =(2x+3)(9 x 2 4x1)+(3 x 3 2 x 2 x)(2) dy dx =(2x+3)(9 x 2 4x1)+(6 x 3 4 x 2 2x)

Practice 1:
Given that y = 4x3 (3x + 1)5, find dy/dx

Solution:
y = 4x(3x + 1)5
dy/dx 
= 4x3. 5(3x + 1)4.3 + (3x + 1)5.12x2
= 60x3 (3x + 1)4 + 12x2 (3x + 1)5
= 12x2 (3x + 1)4 [5x  + (3x + 1)]
= 12x2 (3x + 1)4 (8x  + 1)