This document provides instructions for finding the derivative of the product of two functions using the product rule. It includes examples of taking the derivative of products like (2x+3)(3x-9) and (x^2+3)(4x+1). It then presents the product rule formula and shows how to use it to derive the derivative of a function like Y=(x+3)(4+x).

1. 1
Finding the Derivative of the Product of Two
Functions
Part A
1) Find the derivative of this by multiplying out the brackets:
(i) y = (2x+3) (3x - 9)
(ii) Write down the derivative of u = 2x+3
(iii) Write down the derivative of v = 3x – 9
2) Find the derivative of the following by multiplying out the bracket.
(i) y = (x² + 3)(4x + 1)
(ii) Write down the derivative of y = x² + 3
(iii) Write down the derivative of y = 4x+1
3) Think of your own example of finding the derivative of two straight forward
functions by multiplying out the brackets. Find the derivative of each of these
functions separately.

2. 2
Part B
Complete this table:
Let the first function be called u(x) and the second function called v(x)
First
function 𝒖(𝒙)
Derivative
𝒅
𝒅𝒙
(𝒖( 𝒙))
Second
Function
𝒗(𝒙)
Derivative
𝒅
𝒅𝒙
(𝒗( 𝒙))
Derivative of the
product
𝒅
𝒅𝒙
(𝒖( 𝒙). 𝒗( 𝒙))
𝟐𝒙 + 𝟑 𝟑𝒙 − 𝟗
𝒙 𝟐
+ 𝟑 𝟒𝒙 + 𝟏
𝒖(𝒙) 𝒖′(𝒙) 𝒗(𝒙) 𝒗′(𝒙)
The product rule:
𝒅
𝒅𝒙
(𝒖( 𝒙). 𝒗( 𝒙))=

3. 3
Write down in words a generalization that describes how to find the derivative of the
product of two functions:
𝒚 = 𝒖( 𝒙). 𝒗(𝒙)
Use this generalization to find the derivative of:
Y = (x+3) 4x