3.9 Exponential and Logarithmic Derivatives Mon Nov 9 Do Now Find the derivatives of: 1) 2)
Jan 18, 2018
3.9 Exponential and Logarithmic Derivatives
Mon Nov 9Do Now
Find the derivatives of:
1) 2)
HW Review p.181 #1-35
Exponential + Logarithmic Functions
• Logarithmic and exponential functions are among the most common functions encountered in applications.
• Population curves consist of logarithmic functions, particularly the natural logarithm.
• Growth/Decay, business applications use exponential functions
• Thm- For any constant b > 0,
• Thm- In particular,
Derivative of Natural Log• To determine the derivative of the
natural logarithm, let’s take a look at the graph of lnx and its slopes
Derivative of ln x cont’d
• Thm- For x > 0,
Example:
• Find the derivative of f(x) = x ln x and g(x) = x 10^x
Other Base Logarithms
• We can calculate the derivative of other base logs by using the change-of-base formula using ln x
•
Ex
• Find the derivative of
You try
• Find the derivatives• 1)
• 2)
Logarithmic Differentiation
• Logarithmic Differentiation can be used in place of several product/quotient rules
• Ex:
Logarithmic Differentiation
• 1) Take ln of both sides• 2) Use log rules to separate each factor• 3) Differentiate both sides (chain rule)• 4) Multiply by f(x) (original)
Ex
• Use log differentiation
Ex 2• Differentiate using log dif.
Closure
• Find the derivative using logarithmic differentiation
• HW: p.187 #1-49 odds, 79• 3.7-3.9 Quiz Mon
3.7-3.9 HW/Quiz ReviewTues Nov 10
• Do Now• Find the derivative of each• 1)
• 2)
HW Review: p.187 #1-49 79
3.7-3.9 Review
• Chain Rule– May contain all old rules (product, quotient, trig,
etc)• Derivatives of Inverses
– Explicit Derivatives (switch variables and differentiate)
– Inverse Trig (1 of them)• Logarithmic and Exponential Derivatives
– Most likely be included in chain rule– Logarithmic differentiation technique
Closure
• Journal Entry: How useful is logarithmic differentiation? When would you use it? When wouldn’t you?
• Quiz Thurs up to section 3.9