1. All slides © Christine Crisp
5: The Chain Rule
All slides © Christine Crisp

2. A reminder of the differentiation done so far!
The gradient at a point on a curve is defined as the gradient of the tangent at that point.
The function that gives the gradient of a curve at any point is called the gradient function.
The process of finding the gradient function is called differentiating.
The rules we have developed for differentiating are:

3. Differentiating a function of a function .
Suppose and
Let
We can find by multiplying out the brackets:
However, the chain rule will get us to the answer without needing to do this ( essential if we had, for example, )

4. u is on the inside of the brackets
Consider again and
Let 2 3 ) 4 ( - x
Let
Then,
Differentiating both these expressions:
We must get the letters right.
u is on the inside of the brackets

5. ) 4 ( - x Consider again and . Let Let Then,2 3 ) 4 ( - x
Let
Then,
Differentiating both these expressions:
Now we can substitute for u

6. ) 4 ( - x Consider again and . Let Let Then,2 3 ) 4 ( - x
Let
Then,
Differentiating both these expressions:
Can you see how to get to the answer which we know is
We need to multiply by

7. So, we have
and to get we need to multiply by
So,
This expression is behaving like fractions with the s on the r,h,s, cancelling.

8. So, we have
and to get we need to multiply by
So,
This expression is behaving like fractions with the s on the r,h,s, cancelling.
Although these are not fractions, they come from taking the limit of the gradient, which is a fraction.

9. e.g. 1 Find if
We need to recognise the function as and identify the inner function ( which is u ).
Solution:

10. e.g. 1 Find if
We need to recognise the function as and identify the inner function ( which is u ).
Solution:
Then
Let
Differentiating:
Substitute for u
Tidy up by writing the constant first
We don’t multiply out the brackets

11. e.g. 2 Find if
Solution: We can start in 2 ways. Can you spot them?
Either write
and then let
Always use fractions for indices, not decimals. so
Or, if you don’t notice this, start with
Then

12. e.g. 2 Find if
Solution: Whichever way we start we get
and

13. SUMMARY
The chain rule is used for differentiating functions of a function. If
where , the inner function.

14. Exercise
Use the chain rule to find for the following: 1. 2. 3. 4. 5. 1.
Solutions:

15. Solutions 2. 3.

16. Solutions 4.

17. Solutions 5.

18. Stop here for now

19. The chain rule can also be used to differentiate functions involving e.
e.g. 3. Differentiate
Solution: The inner function is the 1st operation on x, so u = -2x.
Let

20. Exercise
Use the chain rule to differentiate the following: 1. 2. 3.
Solutions: 1.

21. Solutions 2. 3.

22. Later we will want to reverse the chain rule to integrate some functions of a function.
To prepare for this, we need to be able to use the chain rule without writing out all the steps.
e.g. For we know that
The derivative of the inner function which is
has been multiplied by the derivative of the outer function
which is
( I’ve put dashes here because we want to ignore the inner function at this stage. We must not differentiate it again. )

23. So, the chain rule says
differentiate the inner function
multiply by the derivative of the outer function
e.g.

24. So, the chain rule says
differentiate the inner function
multiply by the derivative of the outer function
e.g.
( The inner function is )

25. So, the chain rule says
differentiate the inner function
multiply by the derivative of the outer function
e.g.
( The inner function is )
( The outer function is )

26. So, the chain rule says
differentiate the inner function
multiply by the derivative of the outer function
e.g.
( The inner function is )
( The outer function is )

27. Below are the exercises you have already done using the chain rule with exponential functions.
See if you can get the answers directly. 1. 2. 3.
Answers: 1. 2. 3.
Notice how the indices never change.

28. TIP: When you are practising the chain rule, try to write down the answer before writing out the rule in full. With some functions you will find you can do this easily.
However, be very careful. With some functions it’s easy to make a mistake, so in an exam don’t take chances. It’s probably worth writing out the rule.

30. 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.

31. The chain rule is used for differentiating functions of a function.
where , the inner function. If
SUMMARY

32. e.g. 1 Find if
Solution:
Let
Then
Differentiating:
We need to recognise the function as and identify the inner function ( which is u ).
We don’t multiply out the brackets

33. The chain rule can also be used to differentiate functions involving e.
Let
e.g. Differentiate
Solution: The inner function is the 1st operation on x so here it is -2x.

34. Later we will want to reverse the chain rule to integrate some functions of a function.
To prepare for this, we need to be able to use the chain rule without writing out all the steps.
e.g. For we know that
has been multiplied by the derivative of the outer function
The derivative of the inner function which is
( I’ve put dashes here because we want to ignore the inner function at this stage. We mustn’t differentiate it again. )
which is

35. So, the chain rule says
differentiate the inner function
multiply by the derivative of the outer function
( The outer function is )
( The inner function is )
e.g.
With exponential functions, the index never changes.