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

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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,

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,

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

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

SUMMARY

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.

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)

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

Similarly, is a translation from of

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

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,

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.”

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,

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.”

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

Exercises 3. 4. 5.

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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:

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

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