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Differentiation
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2. The Product and Quotient Rules
2.4
The Product and Quotient Rules
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3. Objectives Find the derivatives of functions using the Product Rule.
Find the derivatives of functions using the Quotient Rule.
Use derivatives to answer questions about real-life situations.

4. The Product Rule

5. The Product Rule
The Product Rule of two functions are as follows:

6. Example 2 – Using the Product Rule
Find the derivative of
Solution:
Rewrite the function. Then use the Product Rule to find the derivative.

7. Example 2 – Solution
cont’d

8. The Product Rule
You now have two differentiation rules that deal with products—the Constant Multiple Rule and the Product Rule. The difference between these two rules is that the Constant Multiple Rule is used when one of the factors is a constant
whereas the Product Rule is used when both of the factors are variable quantities.

9. The Product Rule
The next example compares these two rules.

10. Example 3 – Comparing Differentiation Rules
Find the derivative of each function. a. b.
Solution:
a. Because both factors are variable quantities, use the Product Rule.

11. Example 3 – Solution
cont’d
b. Because one of the factors is a constant, use the Constant Multiple Rule.

12. Example 3 – Solution
cont’d

13. The Product Rule
The Product Rule can be extended to products that have more than two factors. For example, if f, g and h are differentiable functions of x, then

14. The Quotient Rule

15. The Quotient Rule
You have seen that by using the Constant Rule, the Power Rule, the Constant Multiple Rule, and the Sum and Difference Rules, you were able to differentiate any polynomial function.
By combining these rules with the Quotient Rule, you can now differentiate any rational function.

16. The Quotient Rule
From the Quotient Rule, you can see that the derivative of a quotient is not, in general, the quotient of the derivatives.
That is,

17. Example 5 – Finding an Equation of a Tangent Line
Find an equation of the tangent line to the graph of
at x = 1.
Solution:
Apply the Quotient Rule, as shown.

18. Example 5 – Solution
cont’d
When x = 1, the value of the function is y = – 1 and the slope is m = 3.

19. Example 5 – Solution
cont’d
Using the point-slope form of a line, you can find the equation of the tangent line to be
The graph of the function and the tangent line is shown in Figure 2.28.
Figure 2.28

20. The Quotient Rule
Not every quotient needs to be differentiated by the Quotient Rule. For instance, each quotient in the next example can be considered as the product of a constant and a function of x.
In such cases, the Constant Multiple Rule is more efficient than the Quotient Rule.

21. Example 7 – Using the Constant Multiple Rule

22. Application

23. Example 8 – Rate of Change of Systolic Blood Pressure
As blood moves from the heart through the major arteries out to the capillaries and back through the veins, the systolic blood pressure continuously drops. Consider a person whose systolic blood pressure P (in millimeters of mercury) is given by
where t is measured in seconds. At what rate is the blood pressure changing 5 seconds after blood leaves the heart?

24. Example 8 – Solution
Begin by applying the Quotient Rule.

25. Example 8 – Solution When t = 5, the rate of change is
cont’d
When t = 5, the rate of change is
So, the pressure is dropping at a rate of 1.48 millimeters per second at t = 5 seconds.