3.9 Notes pd 7
October 03, 2016
Warm up
1. If f(x) = 3x5, find f'(3) algebraically. Then find the equation of the line tangent
to f(x) at the point (3, f(3)).
h(x)
2. Find y':
g(x)
3. If f(x) = g(h(x)), find f '(2)
4. Use a numeric (table) approach
to find the limit as n approaches infinity of
(1 + 1/n) ^ n
3.9 Notes pd 7
October 03, 2016
Recall: Properties of Logarithms:
a = logbc if and only if b =c, and b ≠1
a
• logb(cd)=dlogbc
• logbcd = logbc + logbd
• logbc/d = logbc-logbd
ln x = logex where e = 2.71828...
Recall: Change of Base:
logbc = (log c)/(log b) or (ln c)/ (ln b)
3.9 Notes pd 7
October 03, 2016
Objectives
• Students will derive exponential and logarithmic functions
• Students will be able to use logarithm properties to simplify functions
• Students will be able to find antiderivatives of functions
5. Use the definition of derivative to determine
d/dx (ln x):
3.9 Notes pd 7
6. Exponential Functions:
Use the definition of the derivative to express
f '(x) when f(x) = 2x.
Lets look at different bases.
f(x) = 3x
f(x) = 4x
f(x) = ex
October 03, 2016
3.9 Notes pd 7
Proof of if f(x) = ex then f ' (x) = ex
October 03, 2016
3.9 Notes pd 7
October 03, 2016
7. Derivatives of an exponential function and the
natural exponential function. Start with u is a
differentiable function of x and b is a constant, then d/
dx (eu)= ____ , d/dx (bx)= ____ , d/dx (bu)= ____
3.9 Notes pd 7
October 03, 2016
Example 1:
Example 2:
Find f '(x):
Find f '(x):
Example 3:
Example 4:
Find f '(x):
Find
3.9 Notes pd 7
8. If u is a differentiable function of x, then
determine d/dx (ln u)
a. Determine f '(x) if f(x) = ln x3
b. Determine dy/dx if y = ln
(cos3x)
October 03, 2016
3. g(x) = 3sin(lnx)
3.9 Notes pd 7
c. h(x) = ln(sinx)
d. find f ''(x) if f(x) = ln x3
October 03, 2016
3.9 Notes pd 7
Antiderivatives:
1. If f '(x) = 2x, the antiderivative f(x) is ____
2. What is the antiderivative of 1/x? ______
3. What is the antiderivative of y = cos 5x?
4. If f '(x) = 2x, determine f(x):
p121 #1
October 03, 2016
3.9 Notes pd 7
Pg 121 # 1 Compound Interest Problem:
October 03, 2016
3.9 Notes pd 7
October 03, 2016