“Teach A Level Maths” Vol. 2: A2 Core Modules 12a: The Quotient Rule © Christine Crisp

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The following are examples of quotients: (b) (a) can be divided out to form a simple function as there is a single polynomial term in the denominator. For (b) we use the quotient rule.

The quotient rule gives us a way of differentiating functions which are divided. The rule is similar to the product rule. where u and v are functions of x. Memory aid:

Or using the dash notation

e.g. 1 Differentiate to find . Solution: and We now need to simplify.

We could simplify the numerator by taking out the common factor x, but it’s easier to multiply out the brackets. We don’t touch the denominator. Multiplying out numerator: Now collect like terms: and factorise: We leave the brackets in the denominator as the factorised form is simpler.

Quotients can always be turned into products. e.g. can be written as However, differentiation is usually more awkward if we do this. In the quotient above, and ( both simple functions ) In the product , and ( v needs the chain rule )

SUMMARY To differentiate a quotient: Check if it is possible to divide out. If so, do it and differentiate each term. Otherwise use the quotient rule: If , where u and v are both functions of x

Exercise Use the quotient rule, where appropriate, to differentiate the following. Try to simplify your answers: 1. 2. 3.

1. Solution: and

2. Solution: and

3. Solution: Divide out:

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The following are examples of quotients: (b) (a) (a) can be divided out to form a simple function as there is a single polynomial term in the denominator. For (b) we use the quotient rule.

SUMMARY Otherwise use the quotient rule: If , where u and v are both functions of x To differentiate a quotient: Check if it is possible to divide out. If so, do it and differentiate each term.

Solution: e.g. 1 Differentiate to find . We now need to simplify.

We could simplify the numerator by taking out the common factor x, but it’s easier to multiply out the brackets. We don’t touch the denominator. Now collect like terms: and factorise: We leave the brackets in the denominator. ( A factorised form is considered to be simpler. ) Multiplying out numerator: