Product Rule
What is the product rule?
- The product rule states that if is the product of two functions and then
-
-
- This is not given on the exam formula sheet, so you need to remember it
-
- This is sometimes written as where and are both functions of
- Then
- where , and
- Then
- For your final answer make sure you match the notation used in the question
How do I know when to use the product rule?
- The product rule is used to differentiate a product of two functions
- This can easily be confused with a 'function of a function' (see the Chain Rule note)
- is a function of a function, “sin of cos of ”
- is a product of two functions, “sin x times cos ”
- This can easily be confused with a 'function of a function' (see the Chain Rule note)
How do I use the product rule?
- To differentiate
- you'll need to make it clear what and are
- Arranging them in a square can help
- you'll need to make it clear what and are
- STEP 1
Identify the two functions, and- Then differentiate each one with respect to to find and
- STEP 2
Obtain by applying the product rule formula- If , then
- Simplify the answer if
- it is straightforward to do so
- or if the question requires a particular form
- In trickier problems chain rule may have to be used when finding and
Exam Tip
- Using and can save time writing
- lay them out in a 2x2 'square' to help keep which is which straight
- For trickier functions chain rule may be required along with product rule
- i.e. either and/or could be a 'function of a function'
- So chain rule needed to find and
Worked example
Find the derivative of .
We'll use form
Identify the functions and
Differentiate those to find and
Put the pieces together using
Expand the brackets