1. Derivatives of exponential and logarithmic functions
Section 3.9

2. If you recall, the number e is important in many
instances of exponential growth:
Find the following important limit using graphs
and/or tables:

3. Derivative of Definition of the derivative!!!
The limit we just figured!
The derivative of this
function is itself!!!

4. Derivative of
Given a positive base that is not one, we can use a property
of logarithms to write in terms of :

5. Derivative of
Substitution!
Imp. Diff.

6. Derivative of COB Formula! First off, how am I able to express in the
following way???
COB Formula!

7. Summary of the New Rules
(keeping in mind the Chain Rule and any variable restrictions)

8. Now we can realize the FULL POWER of the Power Rule……………observe:
Start by writing x with any real power as a power of e…

9. Power Rule for Arbitrary Real Powers
If u is a positive differentiable function of x and n is
any real number, then is a differentiable function
of x, and
The power rule works for not only integers, not only rational numbers, but any real numbers!!!

10. Quality Practice Problems
Find :
Find :
Find :

11. Quality Practice Problems
Find :
Find :

12. Use Logarithmic Differentiation:
Quality Practice Problems
How do we differentiate a function when both the base and exponent contain the variable???
Find :
Use Logarithmic Differentiation:
1. Take the natural logarithm of both sides of the
equation
2. Use the properties of logarithms to simplify the
equation
3. Differentiate (sometimes implicitly!) the
simplified equation

13. Quality Practice Problems
Find :

14. Quality Practice Problems
Find using logarithmic differentiation:
Differentiate:

15. Quality Practice Problems
Find using logarithmic differentiation:
Substitute:

16. Quality Practice Problems
A line with slope m passes through the origin and is tangent
to the graph of What is the value of m?
 What does the graph look like?
The slope of the curve:
The slope of the line:
Now, let’s set them equal…

17. Quality Practice Problems
A line with slope m passes through the origin and is tangent
to the graph of What is the value of m?
 What does the graph look like?
So, our slope: