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 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. If you want to go straight to the examples, click on the box below. where u and v are functions of x. Examples

The Quotient Rule 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, becomesSimplifying Part of the 2 nd term,, is the derivative of but with respect to x not v.

The Quotient Rule 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,

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

The Quotient Rule 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:

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

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.

The Quotient Rule Exercise Use the quotient rule, where appropriate, to differentiate the following. Try to simplify your answers:

The Quotient Rule 1. and Solution:

The Quotient Rule 2. and Solution:

The Quotient Rule 3. and Solution:

The Quotient Rule Solution: 4. Divide out:

The Quotient Rule We can now differentiate the trig function by writing xytan 

The Quotient Rule So, This answer can be simplified: is defined as Also, So,