1. 8: The Product Rule © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules

2. The Product Rule "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Module C3 AQA Edexcel MEI/OCR

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

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

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

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

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

8. The Product Rule, but I’ve changed the order. 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! The standard order is to have constants first, then powers of x and finally bracketed factors. Let Now we can do (b) in the same way. and

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

10. The Product Rule How many ( x  1 ) factors are common? Let Now we can do (b) in the same way. and The common factors. 

11. The Product Rule  Let Now we can do (b) in the same way. and 

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

13. The Product Rule 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.

14. The Product Rule Reminder: A function such as is a product, BUT we don’t need the product rule. However, the product rule will work even though you shouldn’t use it N.B. When we differentiate, a constant factor just “tags along” multiplying the answer to the 2 nd factor.

15. The Product Rule Reminder: A function such as is a product, BUT we don’t need the product rule. When we differentiate, a constant factor just “tags along” multiplying the answer to the 2 nd factor. However, the product rule will work even though you shouldn’t use it so, as before.

16. The Product Rule Exercise Use the product rule, where appropriate, to differentiate the following. Try to simplify your answers by removing common factors: 1. 2. 3. 4.

17. The Product Rule Solutions: 1. Letand Remove common factors: Notice the order within each term: constants, powers of x, then exponentials.

18. The Product Rule Letand Remove common factors: 2. 3. No need for the product rule: just multiply out.

19. The Product Rule 4. Letand Remove common factors: Did you notice that v was a function of a function?

20. Product Rule or Chain Rule? We can now differentiate all of the following: A simple function could be like any of the following: We differentiate them term by term using the 4 rules for The multiplying constants just “tag along”. simple functions, products and compound functions ( functions of a function ).

21. Product Rule or Chain Rule? Decide how you would differentiate each of the following ( but don’t do them ): (a) (b) (c) (d) Product rule Chain rule This is a simple function For products we use the product rule and for functions of a function we use the chain rule. Chain rule

22. Product Rule or Chain Rule? Exercise Decide with a partner how you would differentiate the following ( then do them if you need the practice ): 1. 2. 3. 4. 5. Write C for the Chain rule and P for the Product Rule C C C or P P P

23. Product Rule or Chain Rule? 1. P 2. C Solutions

24. Product Rule or Chain Rule? 3. C 4.P

25. Product Rule or Chain Rule? 5. CEither POr

26. The Product Rule

27. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

28. The Product Rule SUMMARY  Otherwise use the product rule: If where u and v are both functions of x To differentiate a product:  Check if it is possible to multiply out. If so, do it and differentiate each term.

29. The Product Rule Let e.g. and Remove common factors:

30. The Product Rule We can now differentiate all of the following: A simple function could be like any of the following: We differentiate them term by term using the 4 rules for The multiplying constants just “tag along”. simple functions, products and

31. Product Rule or Chain Rule? Decide how you would differentiate each of the following ( but don’t do them ): (a) (b) (c) (d) Product rule Chain rule This is a simple function For products we use the product rule and for functions of a function we use the chain rule. Chain rule

32. Product Rule or Chain Rule? It can be tricky to distinguish between products and functions of a function. If you do have difficulty try the following: Assume you have a product. Then, you must have 2 factors ( at least ). If you can’t then it isn’t a product. Try to split up the function into one function multiplied by another.