Note that in this lesson we show examples, but not any proof
that the formulas for the product
and quotient rules are correct.
- If , we can find the derivative by first multiplying and getting .
We would then have
- There is another approach to this derivative which helps us with more complicated functions. If we think of as being made up of two functions, x2 and x5, we could try to take the derivative of each and multiply those derivatives. We would have 2x and 5x4 which would give us 10x5. Clearly, this is not correct.
But there is a rule for products which does work.
If f(x) = u x v where u and v are functions of x, we have:
Using this formula for our f(x) = x2 x x5, we have the following:
We call the formula
the product rule
, we do not have to multiply this first in order to calculate the derivative. Using the product
rule we have:
Sometimes we can factor
this easily and sometimes not. But we have a formula for the derivative of
Therefore, we can now find the slope
of a tangent line
at, for example, x
= 1. We have:
The Quotient Rule:
- For functions that involve a fraction, we have a rule for quotients.
We can get the derivative of simple fractions such as
This is because we can rewrite this as
- But usually, a fraction that involves expressions in x on both the top (numerator) and bottom (denominator) of the fraction requires more work. The quotient rule is as follows:
In the following problem set, calculate the derivatives of the functions stipulated. For the first three problems, initially multiply (expand) the given expression
and use the power rule. Then use the product
rule and compare. Which method is easier? For the second three problems, use the quotient rule. Simplify each numerator.