Last lesson Chain Rule Today The Product Rule

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Last lesson – Chain Rule Today - The Product Rule Diff functions that are

Last lesson – Chain Rule Today - The Product Rule Diff functions that are products Like © Christine Crisp

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:

The Product Rule Now we can do (b) in the same way. Let and

The Product Rule Now we can do (b) in the same way. Let and v is a function of a function but since the derivative of the inner function is 1, we can ignore it. However, we will meet more complicated functions of a function later! , but I’ve changed the order. The standard order is to have constants first, then powers of x and finally bracketed factors.

The Product Rule Now we can do (b) in the same way. Let and

The Product Rule Now we can do (b) in the same way. Let and Tip: The cross ( multiply! ) acts as a reminder of the product rule! Don’t be tempted to try to multiply out. Think how many terms there will be! There are common factors.

The Product Rule Now we can do (b) in the same way. Let and

The Product Rule Now we can do (b) in the same way. Let and How many = ( x - 1 ) factors are common? The common factors.

The Product Rule Now we can do (b) in the same way. Let and

The Product Rule Now we can do (b) in the same way. Let and =

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 N. B. You may, at first, find it difficult to simplify

The Product Rule N. B. 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 Product Rule Reminder: A function such as is a product, BUT we don’t

The Product Rule Reminder: A function such as is a product, BUT we don’t need the product rule. When we differentiate, a constant factor just “tags along” multiplying the answer to the 2 nd factor. However, the product rule will work even though you shouldn’t use it N. B.

The Product Rule Exercise Use the product rule, where appropriate, to differentiate the following.

The Product Rule Exercise Use the product rule, where appropriate, to differentiate the following. Try to simplify your answers by removing common factors: 1. 2. 3. 4.

The Product Rule Solutions: 1. Let and order within each term: Remove Notice commonthe

The Product Rule Solutions: 1. Let and order within each term: Remove Notice commonthe factors: constants, powers of x, then exponentials.

The Product Rule 2. Let and Remove common factors: 3. No need for the

The Product Rule 2. Let and Remove common factors: 3. No need for the product rule: just multiply out.

The Product Rule 4. Let and Did you notice that v was a function

The Product Rule 4. Let and Did you notice that v was a function of a function? Remove common factors:

Product Rule or Chain Rule? We can now differentiate all of the following: simple

Product Rule or Chain Rule? We can now differentiate all of the following: simple functions, products and compound functions ( functions of a function ). A simple function could be like any of the following: We differentiate them term by term using the 4 rules for The multiplying constants just “tag along”.

Product Rule or Chain Rule? For products we use the product rule and for

Product Rule or Chain Rule? For products we use the product rule and for functions of a function we use the chain rule. Decide how you would differentiate each of the following ( but don’t do them ): (a) Chain rule (b) Product rule (c) This is a simple function (d) Chain rule

Product Rule or Chain Rule? Exercise Decide with a partner how you would differentiate

Product Rule or Chain Rule? Exercise Decide with a partner how you would differentiate the following ( then do them if you need the practice ): Write C for the Chain rule and P for the Product Rule P 1. 3. C 5. 2. 4. C or P C P

Solutions 1. 2. Product Rule or Chain Rule? P C

Solutions 1. 2. Product Rule or Chain Rule? P C

Product Rule or Chain Rule? 3. 4. C P

Product Rule or Chain Rule? 3. 4. C P

Product Rule or Chain Rule? 5. Either C Or P

Product Rule or Chain Rule? 5. Either C Or P