• # Revision:Chain, Product and Quotient

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.

### Proof

Suppose that is a function of , and is a function of .

If , and are corresponding increments in the variables in the variables , and, then

(1)

When , and tend to zero,

so equation (1) becomes

### Examples

If , find .

Let

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:

When

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:

### Proof

Consider .

Then (1)

Let increase by a small amount , with corresponding increases in , and of , and , so that

(2)

Then, subtracting (1) from (2).

Then, dividing by ,

Let .

Then and ,

### Examples

Differentiate

Let and

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.

### Proof

Writing and , then

(1)

and

(2)

Substituting (1) into (2),

Dividing by ,

Let , then , and

If , find .

Let and .

and