1. “Teach A Level Maths” Vol. 2: A2 Core Modules
3: The Product Rule
© Christine Crisp

2. The product rule gives us a way of differentiating functions which are multiplied together.
Consider (a)
and (b)
(a) can be differentiated by multiplying out the brackets . . .
but, we need an easier way of doing (b) since
has 11 terms!
However, doing (a) gives us a clue for a new method.

3. (a)
Multiplying out:
Now suppose we differentiate the 2 functions in without multiplying them out.
Let and
Multiplying these 2 answers does NOT give
BUT

4. (a)
Multiplying out:
Now suppose we differentiate the 2 functions in without multiplying them out.
Let and
and
BUT

5. (a)
Multiplying out:
Now suppose we differentiate the 2 functions in without multiplying them out.
Let and
and
BUT
Adding these gives the answer we want.

6. So, if
e.g.
where u and v are both functions of x,
Multiply out the brackets:
We need to simplify the answer:
Collect like terms:

7. Now we can do (b) in the same way.
Let
and
v is a function of a function but since the derivative of the inner function is 1, we can ignore it. However, we will meet more complicated functions of a function later!
, but I’ve changed the order.
The standard order is to have constants first, then powers of x and finally bracketed factors.

8. Tip: The cross ( multiply! ) acts as a reminder of the product rule!
Now we can do (b) in the same way.
Let
and
Tip: The cross ( multiply! ) acts as a reminder of the product rule!
Don’t be tempted to try to multiply out. Think how many terms there will be!
There are common factors.

9. = Now we can do (b) in the same way. Let and
How many ( x - 1 ) factors are common?
The common factors. =

10. Now we can do (b) in the same way.
Let
and = =

11. SUMMARY
To differentiate a product:
Check if it is possible to multiply out. If so, do it and differentiate each term.
Otherwise use the product rule:
where u and v are both functions of x If
The product rule says:
multiply the 2nd factor by the derivative of the 1st.
Then add the 1st factor multiplied by the derivative of the 2nd.

12. N.B. You may, at first, find it difficult to simplify the answers to look the same as those given in textbooks. Don’t worry about this but keep trying as it gets easier with practice.

13. Exercise
Use the product rule, where appropriate, to differentiate the following. Try to simplify your answers by removing common factors: 1. 2.

14. 1.
No need for the product rule: just multiply out.

15. 2.
Let
and
Did you notice that v was a function of a function?
Remove common factors:

16. Lets try some more 

17. k(x) = 4x3(5x – 3)
y = (3x2 – x – 7)(7 + 3x)