3.9 Exponential and
Logarithmic Derivatives
Thurs Oct 8
Do Now
Find the derivatives of:
1) f (x) = sinsinsinsin3x
( x+2)1/2
2) g(x) = 4e
HW Review p.181
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,
d
dx
b = b lnb
x
x
• Thm- In particular,
d
dx
e = e lne = e
x
x
x
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,
d
dx
ln x =
1
x
Example:
• Find the derivative of f(x) = x ln x and
g(x) = x 10^x
f (x) = x ln x
g(x) = x× 10
f ¢(x) = (1)ln x + x( 1x )
f ¢(x) = (1)× 10 x + x× 10 x ln10
= ln x + 1
x
Other Base Logarithms
• We can calculate the derivative of other
base logs by using the change-of-base
formula using ln x
•
ln x
log b x =
lnb
Ex
• Find the derivative of y = log10 x
Logarithmic Differentiation
• Logarithmic Differentiation can be used
in place of several product/quotient
2
2
rules
(x +1) (2x - 3)
f (x) =
• Ex:
2
x +1
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
x(x +1)
• Use log differentiation f (x) =
2
(3x -1)
3
Ex 2
• Differentiate using log dif.
f (x) = (x +1)(x + 2)(x + 3)
3
4
5
Closure
• Find the derivative using logarithmic
differentiation
y = (2x +1)(4x ) x - 9
2
• HW: p.187 #3 13 19 23 27 47 59 77
• 3.7-3.10 Test soon (Thurs?)
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