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Chain Rule (Cambridge O Level Additional Maths)
Revision Note
Chain Rule
What is the chain rule?
- The chain rule states if is a function of and is a function of then
- In function notation this could be written
How do I know when to use the chain rule?
- The chain rule is used when we are trying to differentiate composite functions
- “function of a function”
- these can be identified as the variable (usually) does not ‘appear alone’
- – not a composite function, ‘appears alone’
- is a composite function; is tripled and has 2 added to it before the sine function is applied
How do I use the chain rule?
STEP 1
Identify the two functions
Rewrite as a function of;
Write as a function of;
STEP 2
Differentiate with respect to to get
Differentiate with respect to to get
STEP 3
Obtain by applying the formula and substitute back in for
-
In trickier problems chain rule may have to be applied more than once
How do I differentiate (ax + b)n?
- For n = 2 you will most likely expand the brackets and differentiate each term separately
- If n > 2 this becomes time-consuming and if n is not a positive integer we need a different method completely
- The chain rule allows us to use substitution to differentiate any function in the form y = (ax + b)n
- Let u = ax + b, then y = un
- Differentiate both parts separately
- and
- Put both parts into the chain rule
- Substitute u = ax + b back into your answer
How do I differentiate √(ax+b)?
- The chain rule allows us to use substitution to differentiate any function in the form
- Rewrite
- Let u = ax + b, then y = u½
- Differentiate both parts separately
- and
- Put both parts into the chain rule
- Substitute u = ax + b back into your answer
- This method can be used for any fractional power of any linear or non-linear expression
- Provided you know how to differentiate the non-linear expression
Are there any standard results for using chain rule?
- The following general results are particularly useful
- If then
-
- If then
- If then
- If then
- If then
- If then
Examiner Tip
- You should aim to be able to spot and carry out the chain rule mentally (rather than use substitution)
- every time you use it, say it to yourself in your head
“differentiate the first function ignoring the second, then multiply by the derivative of the second function"
- every time you use it, say it to yourself in your head
Worked example
a)
Find the derivative of.
STEP 1 Identify the two functions and rewrite
STEP 2 Find and
STEP 3 Apply the chain rule,
Substitute in terms of back in
b)
Find the derivative of.
(In this solution, we will be applying the mental method discussed in the Exam Tip above)
"... differentiate , ignore "
"... multiply by the derivative of ": differentiate using the result "if , then "
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