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.

(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

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

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

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:

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.

Tip: The cross ( multiply! ) acts as a reminder of 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.

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

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

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

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.

Let and Did you notice that v was a function of a function? Remove common factors: In a differentiation exam, if it is a skills questions then you do not need to simplify but in a merit/excellence question you may need to simplify to solve an equation.