# 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:

Solution:
$\begin{array}{l}y=\left(2x+3\right)\left(3{x}^{3}-2{x}^{2}-x\right)\\ \frac{dy}{dx}=\left(2x+3\right)\left(9{x}^{2}-4x-1\right)+\left(3{x}^{3}-2{x}^{2}-x\right)\left(2\right)\\ \frac{dy}{dx}=\left(2x+3\right)\left(9{x}^{2}-4x-1\right)+\left(6{x}^{3}-4{x}^{2}-2x\right)\end{array}$

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)