1. The Chain Rule

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
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. ) 4 ( - x Consider again Let Then,2 3 ) 4 ( - x
Consider again
Let
Then,
Differentiating both these expressions:
We must get the letters right.

5. Let
Then,
Differentiating both these expressions:
Now we can substitute for u

6. 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
Solution:

10. e.g. 1 Find if
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

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. 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.