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Product rule to be learnt! Chain rule: Quotient rule to be learnt!
The chain rule is used to differentiate composite
If , then . functions (i.e. functions of a function).
It is used when there is a product of 2 functions. The chain rule operates by making the substitution, It is used for differentiating quotients (i.e. fractions).
Example: Differentiate Example: Differentiate
This is a product of 2 functions: .
Using the product rule formula: Differentiate:
So using the chain rule: So,
Factorise top in order to simplify:
Remember these results: Finding the derivative when x = f(y):
We use the result: .
Example: If , find in terms of y.
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Connected rates of change: Example:
The word rate is associated with differentiation. The volume of a sphere is decreasing at a rate of
Key steps: We want .
20cm3/s. Find the rate at which the radius is
(1) Interpret mathematically the information given in the increasing when the radius is 3 cm.
(2) Write down the relevant formula connecting the Using chain rule:
variables and differentiate it. Solution: From question:
(3) Decide on what quantity needs to be calculated.…read more