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Techniques of Differentiation (DP IB Maths: AI HL)
Revision Note
Chain Rule
What is the chain rule?
- The chain rule states if is a function of and is a function of then
-
- This is given in the formula booklet
- 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
Are there any standard results for using chain rule?
- There are five general results that can be useful
- 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.
b)
Find the derivative of.
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Product Rule
What is the product rule?
- The product rule states if is the product of two functions and then
-
- This is given in the formula booklet
- In function notation this could be written as
- ‘Dash notation’ may be used as a shorter way of writing the rule
- Final answers should match the notation used throughout the question
How do I know when to use the product rule?
- The product rule is used when we are trying to differentiate the product of two functions
- these can easily be confused with composite functions (see chain rule)
- is a composite function, “sin of cos of ”
- is a product, “sin x times cos ”
- these can easily be confused with composite functions (see chain rule)
How do I use the product rule?
- Make it clear what and are
- arranging them in a square can help
- opposite diagonals match up
- arranging them in a square can help
STEP 1
Identify the two functions, and
Differentiate both and with respect to to find and
STEP 2
Obtain by applying the product rule formula
Simplify the answer if straightforward to do so or if the question requires a particular form
- In trickier problems chain rule may have to be used when finding and
Examiner Tip
- Use and for the elements of product rule
- lay them out in a 'square' (imagine a 2x2 grid)
- those that are paired together are then on opposite diagonals ( and , and )
- For trickier functions chain rule may be required inside product rule
- i.e. chain rule may be needed to differentiate and
Worked example
a) Find the derivative of.
b) Find the derivative of.
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Quotient Rule
What is the quotient rule?
- The quotient rule states if is the quotient then
-
- This is given in the formula booklet
- In function notation this could be written
- As with product rule, ‘dash notation’ may be used
- Final answers should match the notation used throughout the question
How do I know when to use the quotient rule?
- The quotient rule is used when trying to differentiate a fraction where both the numerator and denominator are functions of
- if the numerator is a constant, negative powers can be used
- if the denominator is a constant, treat it as a factor of the expression
How do I use the quotient rule?
- Make it clear what and are
- arranging them in a square can help
- opposite diagonals match up (like they do for product rule)
- arranging them in a square can help
STEP 1
Identify the two functions, and
Differentiate both and with respect to to find and
STEP 2
Obtain by applying the quotient rule formula
Be careful using the formula – because of the minus sign in the numerator, the order of the functions is important
Simplify the answer if straightforward or if the question requires a particular form
- In trickier problems chain rule may have to be used when finding and,
Examiner Tip
- Use and for the elements of quotient rule
- lay them out in a 'square' (imagine a 2x2 grid)
- those that are paired together are then on opposite diagonals ( and , and )
- Look out for functions of the form
- These can be differentiated using a combination of chain rule and product rule
(it would be good practice to try!) - ... but it can also be seen as a quotient rule question in disguise
- ... and vice versa!
- A quotient could be seen as a product by rewriting the denominator as
- These can be differentiated using a combination of chain rule and product rule
Worked example
Differentiate with respect to .
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