TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Chain, Product and Quotient
The Chain Rule
The chain rule is very important in differential calculus and states that:
This rule allows us to differentiate a vast range of functions.
Suppose that is a function of , and is a function of .
If , and are corresponding increments in the variables in the variables , and, then
When , and tend to zero,
so equation (1) becomes
If , find .
By the Chain Rule, .
In examples such as the above one, with practice it should be possible for you to be able to simply write down the answer without having to let etc. This is because:
In other words, the differential of something in a bracket raised to the power of n is the differential of the bracket, multiplied by multiplied by the contents of the bracket raised to the power of .
The Product Rule
This is another very useful formula, when we have two functions and , multiplied together:
Let increase by a small amount , with corresponding increases in , and of , and , so that
Then, subtracting (1) from (2).
Then, dividing by ,
Then and ,
Again, with practice you shouldn't have to write out and every time.
The Quotient Rule
This formula lets us differentiate two functions divided by each other.
Writing and , then
Substituting (1) into (2),
Dividing by ,
Let , then , and
If , find .
Let and .