1. The Derivatives of ax and logax
OBJECTIVE
Differentiate functions involving ax.
Differentiate functions involving logax.
Find the future value and the rate of growth of an annuity.

2. 3.5 The Derivatives of ax and logax
THEOREM 12

3. 3.5 The Derivatives of ax and logax
Example 1: Differentiate:

4. 3.5 The Derivatives of ax and logax
Quick Check 1
Differentiate:
a.)
b.)
c.)

5. 3.5 The Derivatives of ax and logax
THEOREM 14

6. 3.5 The Derivatives of ax and logax
Example 2: Differentiate:

7. 3.5 The Derivatives of ax and logax
Example 2 (concluded):

8. 3.5 The Derivatives of ax and logax
Quick Check 2
Differentiate:
a.)
b.)
c.)
d.)

9. 3.5 The Derivatives of ax and logax
The value of an annuity is given by
where p is the amount of the regular payment, r the annual percentage rate, and n the number of compounding periods per year.

10. 3.5 The Derivatives of ax and logax
Example 3: Javier deposits $150 every month into an annuity with an annual interest rate of 4.5%. Assume that interest is compounded monthly.
a) Find a function A(t) that gives the value of Javier’s annuity after t years.
b) What is the value of Javier’s annuity after 5 years?
c) What is the rate of change in the value of Javier’s annuity after 5 years?

11. 3.5 The Derivatives of ax and logax
Example 3 (continued):
a) We have p = 150, r = 0.045, and n = 12.

12. 3.5 The Derivatives of ax and logax
Example 3 (continued):
b) In 5 yr, the value of Javier’s annuity will be.

13. 3.5 The Derivatives of ax and logax
Example 3 (continued):
To find the rate of change in the value of Javier’s annuity, we first find the derivative of A(t):

14. 3.5 The Derivatives of ax and logax
Example 3 (concluded):
After 5 yr, the value of Javier’s annuity is changing at the rate of

15. 3.5 The Derivatives of ax and logax
Section Summary
The following rules apply when we differentiate exponential and logarithmic functions whose bases are positive but not the number e:

16. 3.5 The Derivatives of ax and logax
Section Summary
The value of an annuity is given by
where p is the amount of the regular payment, r the annual percentage rate, and n the number of compounding periods per year.