1. “Teach A Level Maths” Vol. 2: A2 Core Modules
6: Differentiating
© Christine Crisp

2. Module C3
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3. x and y have swapped from their usual places.
means
A log is just an index, so
So, to differentiate we just need to differentiate
We have to be careful with the letters:
x and y have swapped from their usual places.
So,

4. x and y have swapped from their usual places.
means
A log is just an index, so
So, to differentiate we just need to differentiate
We have to be careful with the letters:
x and y have swapped from their usual places.
So,

5. x and y have swapped from their usual places.
means
A log is just an index, so
So, to differentiate we just need to differentiate
We have to be careful with the letters:
x and y have swapped from their usual places.
So,
However, for , we want not

6. We’ve already seen that behaves like a fraction,
Compare this with
So,
Hence,
Finally, for , we want the answer in terms of x.
So, since we get

7. SUMMARY

8. Compound Functions Involving logs
We can always use the chain rule to differentiate compound log functions.
However, the first 3 log laws met in AS can simplify the work.
Using ln instead of log these are:
It’s important to use these laws as they change compound functions into simple ones.

9. e.g.1 Use log laws to simplify the following and hence find .
(b)
(c)
Solution:
(a)
is a constant so its derivative is zero
(b)
(c)

10. It may seem surprising that the gradient functions of
are all given by
The graphs show us why:
is a translation from of
. So, we have

11. Similarly, is a translation fromof

12. Since the graphs are translations parallel to the y-axis, the gradients are the same.

13. e.g.2 Differentiate with respect to x.
Solution:
(a) cannot be simplified. There’s no rule for the log of a sum or difference.
We must use the chain rule.
So,

14. So,
We can generalise this to get a really useful result
where is the derivative of the function
This rule in words is:
“ The derivative of the inner function divided by the inner function.”

15. e.g.3 Differentiate
Solution:
The brackets are essential here.
We can now differentiate each term separately, using the result from the last example:
So,

16. SUMMARY
To differentiate compound log functions,
Use the log laws to simplify the expression if possible.
Differentiate each term using
This rule in words is:
“ The derivative of the inner function divided by the inner function.”

17. Exercises
Differentiate the following with respect to x: 1. 2. 3. 4. 5.
Solutions: 1. 2.

18. Exercises 3. 4. 5.

20. 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.

21. To differentiate compound log functions,
Use the log laws to simplify the expression if possible.
Differentiate each term using
“ The derivative of the inner function divided by the inner function.”
This rule in words is:

22. e.g.1 Use log laws to simplify the following and hence find .
Solution:
(a)
(b)
(c)

23. Solution:
e.g. Differentiate
We can now differentiate each term separately, using:
So,