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

Differentiating "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

Differentiating 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,

Differentiating 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,

Differentiating 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. However, for, we want not So,

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

SUMMARY

Differentiating 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.

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

Differentiating 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

Differentiating Similarly, is a translation from of

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

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

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

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

Differentiating SUMMARY 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:

Differentiating Exercises Differentiate the following with respect to x : Solutions: 2.

Differentiating Exercises

Differentiating

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Differentiating 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:

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

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