So far, we can differentiate power functions (
), exponential functions (
), and the two fundamental trigonometric functions (
and
). With the sum rule and constant multiple rules, we can also compute the derivative of combined functions.
Example 2.3.1.
Differentiate
Because is a sum of basic functions, we can now quickly say that
What about a product or quotient of two basic functions, such as
or
While the derivative of a sum is the sum of the derivatives, it turns out that the rules for computing derivatives of products and quotients are more complicated.
Preview Activity 2.3.1.
Subsection 2.3.1 The product rule
As part (b) of
Preview Activity 2.3.1 shows, it is not true in general that the derivative of a product of two functions is the product of the derivatives of those functions. To see why this is the case, we consider an example involving meaningful functions.
Say that an investor is regularly purchasing stock in a particular company. Let represent the number of shares owned on day where represents the first day on which shares were purchased. Let give the value of one share of the stock on day note that the units on are dollars per share. To compute the total value of the stock on day we take the product
Observe that over time, both the number of shares and the value of a given share will vary. The derivative measures the rate at which the number of shares is changing, while measures the rate at which the value per share is changing. How do these respective rates of change affect the rate of change of the total value function?
To help us understand the relationship among changes in and let’s consider some specific data.
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Suppose that on day 100, the investor owns 520 shares of stock and the stock’s current value is $27.50 per share. This tells us that and
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On day 100, the investor purchases an additional 12 shares (so the number of shares held is rising at a rate of 12 shares per day).
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On that same day the price of the stock is rising at a rate of 0.75 dollars per share per day.
In calculus notation, the latter two facts tell us that
(shares per day) and
(dollars per share per day). At what rate is the value of the investor’s total holdings changing on day 100?
Observe that the increase in total value comes from two sources: the growing number of shares, and the rising value of each share. If only the number of shares is increasing (and the value of each share is constant), the rate at which which total value would rise is the product of the current value of the shares and the rate at which the number of shares is changing. That is, the rate at which total value would change is given by
Note particularly how the units make sense and show the rate at which the total value
is changing, measured in dollars per day.
If instead the number of shares is constant, but the value of each share is rising, the rate at which the total value would rise is the product of the number of shares and the rate of change of share value. The total value is rising at a rate of
Of course, when both the number of shares and the value of each share are changing, we have to include both of these sources. In that case the rate at which the total value is rising is
We expect the total value of the investor’s holdings to rise by about $720 on the 100th day.
Next, we expand our perspective from the specific example above to the more general and abstract setting of a product
of two differentiable functions,
and
If
our work above suggests that
Indeed, a formal proof using the limit definition of the derivative can be given to show that the following rule, called the
product rule, holds in general.
Product Rule.
If and are differentiable functions, then their product is also a differentiable function, and
In light of the earlier example involving shares of stock, the product rule also makes sense intuitively: the rate of change of
should take into account both how fast
and
are changing, as well as how large
and
are at the point of interest. In words the product rule says: if
is the product of two functions
(the first function) and
(the second), then “the derivative of
is the first times the derivative of the second, plus the second times the derivative of the first.” It is often a helpful mental exercise to say this phrasing aloud when executing the product rule.
Example 2.3.2.
If we can use the product rule to differentiate The first function is and the second function is By the product rule, will be given by the first, times the derivative of the second, plus the second, times the derivative of the first, That is,
Activity 2.3.2.
Use the product rule to answer each of the questions below. Throughout, be sure to carefully label any derivative you find by name. It is not necessary to algebraically simplify any of the derivatives you compute.
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-
-
Determine the slope of the tangent line to the curve
at the point
if
is given by the rule
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Find the tangent line approximation
to the function
at the point
if
is given by the rule
Subsection 2.3.2 The quotient rule
Because quotients and products are closely linked, we can use the product rule to understand how to take the derivative of a quotient. Let be defined by where and are both differentiable functions. It turns out that is differentiable everywhere that We would like a formula for in terms of and Multiplying both sides of the formula by we observe that
Now we can use the product rule to differentiate
We want to know a formula for so we solve this equation for
and dividing both sides by we have
Finally, we recall that Substituting this expression in the preceding equation, we have
This calculation gives us the
quotient rule.
Quotient Rule.
If and are differentiable functions, then their quotient is also a differentiable function for all where and
As with the product rule, it can be helpful to think of the quotient rule verbally. If a function
is the quotient of a top function
and a bottom function
then
is given by “the bottom times the derivative of the top, minus the top times the derivative of the bottom, all over the bottom squared.”
Example 2.3.3.
If we call the top function and the bottom function. By the quotient rule, is given by the bottom, times the derivative of the top, minus the top, times the derivative of the bottom, all over the bottom squared, That is,
In this particular example, it is possible to simplify by removing a factor of from both the numerator and denominator, so that
In general, we must be careful in doing any such simplification, as we don’t want to execute the quotient rule correctly but then make an algebra error.
Activity 2.3.3.
Use the quotient rule to answer each of the questions below. Throughout, be sure to carefully label any derivative you find by name. That is, if you’re given a formula for clearly label the formula you find for It is not necessary to algebraically simplify any of the derivatives you compute.
-
-
-
Determine the slope of the tangent line to the curve
at the point where
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When a camera flashes, the intensity of light seen by the eye is given by the function
where is measured in candles and is measured in milliseconds. Compute and include appropriate units on each value; and discuss the meaning of each.