1. Derivatives of Exponential and Logarithmic Functions
AP CALCULUS AB
Chapter 3:
Derivatives
Section 3.9:
Derivatives of Exponential and Logarithmic Functions

2. What you’ll learn about
Derivative of ex
Derivative of ax
Derivative of ln x
Derivative of logax
Power Rule for Arbitrary Real Powers
… and why
The relationship between exponential and logarithmic functions provides a powerful differentiation tool called logarithmic differentiation.

3. Derivative of ex

4. Example Derivative of ex

5. Derivative of ax

6. Derivative of ln x

7. Example Derivative of ln x

8. Derivative of logax

9. Rule 10 Power Rule For Arbitrary Real Powers

10. Example Power Rule For Arbitrary Real Powers

11. Logarithmic Differentiation
Sometimes the properties of logarithms can be used to simplify the differentiation process, even if logarithms themselves must be introduced as a step in the process.
The process of introducing logarithms before differentiating is called logarithmic differentiation.

12. Example Logarithmic Differentiation

13. In Review
Rule Example

14. Section 3.9 – Derivatives of Exponential and Logarithmic Functions
Derivation of this derivative:

15. Let’s use these formulas!
Using find
Using find

16. At what point on the graph of the function y=3t – 4 does the tangent line have slope 12?
How can we find this out?
Find dy/dt and set it equal to zero.
Solve for t.
Evaluate f(t) in the original equation.
Hint: t is a messy number, store value for accuracy in final answer.

17. A line with slope m=1/a passes through the origin and is tangent to the graph of y = ln x. What is the value of m?
We know
Point of tangency has coordinates (a, ln a) for some positive a
Tangent line has slope m = 1/a
Tangent line passes through the origin so equation has point (0, 0)
Find slope with formula and set it equal to 1/a. Solve for a.

18. Sometimes we should use logarithms to simplify before differentiation.
Find dy/dx for y = x x
Use logs to lose the exponent, THEN take the derivative implicitly.

19. How Fast does H1N1 Spread?
The spread of a flu in a certain school is modeled by the equation
Where P(t) is the total number of students infected t days after the flu
was first noticed. Many of them may already be well again at time t.
Estimate the initial number of students infected with the flu.
How fast is the flu spreading after 3 days?
When will the flu spread at its maximum rate? What is this rate?