The Product Rule The Product Rule The product

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The Product Rule

The Product Rule

The Product Rule The product rule gives us a way of differentiating functions which

The Product Rule The product rule gives us a way of differentiating functions which are multiplied together. Consider (a) and (b) (a) can be differentiated by multiplying out the brackets. . . but, we need an easier way of doing (b) since has 11 terms! However, doing (a) gives us a clue for a new method.

The Product Rule (a) Multiplying out: Now suppose we differentiate the 2 functions in

The Product Rule (a) Multiplying out: Now suppose we differentiate the 2 functions in without multiplying them out. Let and Multiplying these 2 answers does NOT give BUT

The Product Rule (a) Multiplying out: Now suppose we differentiate the 2 functions in

The Product Rule (a) Multiplying out: Now suppose we differentiate the 2 functions in without multiplying them out. Let BUT and

The Product Rule (a) Multiplying out: Now suppose we differentiate the 2 functions in

The Product Rule (a) Multiplying out: Now suppose we differentiate the 2 functions in without multiplying them out. Let BUT and Adding these gives the answer we want.

The Product Rule So, if e. g. where u and v are both functions

The Product Rule So, if e. g. where u and v are both functions of x, Multiply out the brackets: We need to simplify the answer: Collect like terms:

SUMMARY The Product Rule To differentiate a product: Ø Check if it is possible

SUMMARY The Product Rule To differentiate a product: Ø Check if it is possible to multiply out. If so, do it and differentiate each term. Ø Otherwise use the product rule: where u and v are both functions of x If The product rule says: • multiply the 2 nd factor by the derivative of the 1 st. • Then add the 1 st factor multiplied by the derivative of the 2 nd.

The Product Rule NOTE: You may, at first, find it difficult to simplify the

The Product Rule NOTE: You may, at first, find it difficult to simplify the answers to look the same as those given in textbooks. Don’t worry about this but keep trying as it gets easier with practice.

The Quotient Rule

The Quotient Rule

The Quotient Rule The following are examples of quotients: (a) (b) (c) (d) (c)

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

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. This rule can be derived from the product rule but it is complicated. Therefore, let’s go straight to the examples.

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

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

The Quotient Rule We could simplify the numerator by taking out the common factor

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. Multiplying out numerator: Now collect like terms: and factor: We leave the brackets in the denominator as the factorised form is simpler.

SUMMARY The Quotient Rule To differentiate a quotient: Ø Check if it is possible

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

The Quotient Rule Solution: Divide out:

The Quotient Rule Solution: Divide out:

The Quotient Rule Solution: and

The Quotient Rule Solution: and

The Quotient Rule SUMMARY To differentiate a quotient: Ø Check if it is possible

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

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