The Quotient Rule

The following are examples of quotients: (a) (b) (c) (d) (c) can be divided out to form a simple function as there is a single polynomial term in the denominator. For the others 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. This rule can be derived from the product rule but it is complicated. where u and v are functions of x.

We can develop the quotient rule by using the product rule! The problem now is that this v is not the same as the v of the product rule. That v is replaced by. So, becomes Simplifying Part of the 2 nd term,, is the derivative of but with respect to x not v.

We use the chain rule: So, Make the denominators the same by multiplying the numerator and denominator of the 1 st term by v. Write with a common denominator: Then,

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

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 as the factorised form is simpler. Multiplying out numerator:

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.

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

1. and Solution:

2. Divide out: