5.5 Differentiation of Logarithmic Functions

1
5.5 Differentiation of Logarithmic Functions

• By
• Dr. Julia Arnold and Ms. Karen Overman
• using Tans 5th edition Applied Calculus for the
  managerial , life, and social sciences text

2
Now we will find derivatives of logarithmic
functions and we will Need rules for finding
their derivatives. Rule 3 Derivative of ln x
Lets see if we can discover why the rule is as
above.
First define the natural log function as follows
Now rewrite in exponential form
Now differentiate implicitly
3
Example 1 Find the derivative of f(x) xlnx.
Solution This derivative will require the
product rule.
Product Rule (1st)(derivative of 2nd)
(2nd)(derivative of 1st)
4
Example 2 Find the derivative of g(x) lnx/x
Solution This derivative will require the
quotient rule.
Quotient Rule (bottom)(derivative of top)
(top)(derivative of bottom)
(bottom)²
5
Why dont you try one Find the derivative of y
x²lnx .
The derivative will require you to use the
product rule. Which of the following is the
correct?
y 2
y 2xlnx
y x 2xlnx
6
No, sorry that is not the correct answer. Keep
in mind -
Product Rule (1st)(derivative of 2nd)
(2nd)(derivative of 1st)
Try again. Return to previous slide.
7
Good work! Using the product rule
F(x) (1st)(derivative of 2nd)
(2nd)(derivative of 1st)
y x² (lnx)(2x) y x
2xlnx This can also be written y x(12lnx)
8
Here is the second rule for differentiating
logarithmic functions.
Rule 4 The Chain Rule for Log Functions
In words, the derivative of the natural log of
f(x) is 1 over f(x) times the derivative of
f(x) Or, the derivative of the natural log of
f(x) is the derivative of f(x) over f(x)
9
Example 3 Find the derivative of
Solution Using the chain rule for logarithmic
functions.
Derivative of the inside, x²1
The inside, x²1
10
Example 4 Differentiate
Solution There are two ways to do this problem.
One is easy and the other is more difficult.
The difficult way
11
The easy way requires that we simplify the log
using some of the expansion properties.
Now using the simplified version of y we find y .
12
Now that you have a common denominator, combine
into a single fraction.
Youll notice this is the same as the first
solution.
13
Example 6 Differentiate
Solution Using what we learned in the previous
example. Expand first
Now differentiate
Recall lnexx
14
Find the derivative of .
Following the method of the previous two
examples. What is the next step?
15
This method of differentiating is valid, but it
is the more difficult way to find the derivative.
It would be simplier to expand first using
properties of logs and then find the derivative.
Click and you will see the correct expansion
followed by the derivative.
16
Correct. First you should expand to
Then find the derivative using the rule 4 on each
logarithm.
Now get a common denominator and simplify.
17
Example 7 Differentiate
Solution Although this problem could be easily
done by multiplying the expression out, I would
like to introduce to you a technique which you
can use when the expression is a lot more
complicated. Step 1 Take the ln of both sides.
Step 2 Expand the complicated side.
Step 3 Differentiate both side (implicitly for
ln y )
18
Step 4 Solve for y .
Step 5Substitute y in the above equation and
simplify.
19
Continue to simplify
20
Lets double check to make sure that derivative
is correct by Multiplying out the original and
then taking the derivative.
Remember this problem was to practice the
technique. You would not use it on something
this simple.
21
Consider the function y xx.
Not a power function nor an exponential function.
This is the graph domain x gt 0
What is that minimum point?
Recall to find a minimum, we need to find the
first derivative, find the critical numbers and
use either the First Derivative Test or the
Second Derivative Test to determine the extrema.
22
To find the derivative of y xx , we will take
the ln of both sides first and then expand.
Now, to find the derivative we differentiate both
sides implicitly.
23
To find the critical numbers, set y 0 and
solve for x.
Now test x 0.1 in y, y(0.1) -1.034 lt 0
and x 0.5 in y, y(0.5) 0.216 gt 0
Thus, the minimum point occurs at x 1/e or
about .37
24
We learned two rules for differentiating
logarithmic functions
Rule 3 Derivative of ln x
Rule 4 The Chain Rule for Log Functions
We also learned it can be beneficial to expand a
logarithm before you take the derivative and
that sometimes it is useful to take the natural
log (ln) of both sides of an equation, rewrite
and then take the derivative implicitly.