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5.1 The Natural Logarithmic Function: Differentiation
After this lesson, you should be able to:
Develop and use properties of the natural logarithmic function.Understand the definition of the number e.Find derivatives of functions involving the natural logarithmic function.
Now we will find derivatives of logarithmic functions and we will need rules for finding their derivatives.
Derivative of ln x 0x
Let’s see if we can discover why the rule is as above.
xy lnFirst define the natural log function as follows:
xe y
Now differentiate implicitly:
xey
ye
y
y
11
1
Now rewrite in exponential form:
xx
dx
d 1ln
Theorem 5.1 Properties of the Natural Logarithmic Function
Example 1 Find the derivative of f(x)= xlnx.
Solution This derivative will require the product rule.
1ln1
)('
ln)(
xxxxf
xxxf
xxf ln1)(
Product Rule:
(1st)(derivative of 2nd) + (2nd)(derivative of 1st)
Example 2 Find the derivative of g(x)= lnx/x
Solution This derivative will require the quotient rule.
2
1ln1
)(
ln)(
x
xxx
xg
x
xxg
Quotient Rule:
(bottom)(derivative of top) – (top)(derivative of bottom)
(bottom)²
2
ln1)(
x
xxg
Why don’t 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
No, sorry that is not the correct answer.
Keep in mind - Product Rule:
(1st)(derivative of 2nd) + (2nd)(derivative of 1st)
Try again.
F ’(x) = (1st)(derivative of 2nd) + (2nd)(derivative of 1st)
Good work! Using the product rule:
y ’ = x² + (lnx)(2x)
y ’ = x + 2xlnx
This can also be written y’ = x(1+2lnx)
x
1
Theorem 5.2 Logarithmic Properties
The Chain Rule for Log Functions
)(
)()(ln
xf
xfxf
dx
d 0)( xf
Here is the second rule for differentiating logarithmic 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)
Theorem 5.3 Derivative of the Natural Logarithmic Function
Example 3 Find the derivative of )1ln()( 2 xxf
Solution Using the chain rule for logarithmic functions.
1
2)(
)1ln()(
2
2
x
xxf
xxf
Derivative of the inside, x²+1
The inside, x²+1
Example 4 Differentiate 632 )2)(1(ln xxy
Solution There are two ways to do this problem. One is easy and the other is more difficult. The difficult
way:
21
41820
21
29102
21
2992
21
21922
21
222118
21
223261
21
21
32
24
7322
3
32
33
632
3253
632
635322
632
632532
632
632
xx
xxx
xx
xxx
xx
xxxx
xx
xxxxx
xx
xxxxx
xx
xxxxx
xx
xxdxd
y
The easy way requires that we simplify the log using some of the expansion properties.
2ln61ln2ln1ln)2)(1(ln 32632632 xxxxxxy
Now using the simplified version of y we find y ’.
2
36
1
2
2ln61ln
3
2
2
32
x
x
x
xy
xxy
632 )2)(1(ln xxy
12
136
21
2223
22
32
3
xx
xx
xx
xxy
Now that you have a common denominator, combine into a singlefraction.
21
41820
12
1818
21
42
12
136
21
22
32
24
23
24
32
4
23
22
32
3
xx
xxx
xx
xx
xx
xx
xx
xx
xx
xxy
Now we can simplify
You’ll notice this is the same as the first solution.
Example 5 Differentiate )ln(22 tettg
222 ln2lnln)ln()(22
ttetettg tt
Solution Using what we learned in the previous example.Expand first:
Now differentiate:
tt
tg
tttg
22
)(
ln2)( 2
Recall lnex=x
Now you try: Find the derivative of .
3
4lnx
xy
Following the method of the previous two examples.What is the next step?
3xln4xlnytoExpanding
3x
4x
dx
d
3-x
4x1
y'toatingDifferenti
This method of differentiating is valid, but it is the more difficult way to find the derivative.
It would be simpler to expand first using properties of logs and then find the derivative.
Click and you will see the correct expansion followed by the derivative.
Correct. First you should expand to 3ln4ln xxy
Then find the derivative using the rule 4 on each logarithm.
3
1
4
1
xxy'
Now get a common denominator and simplify.
34
7
34
4
34
3
xxy'
xx
x
xx
xy'
Example 6 Differentiate )1)(1( 2 xxxy
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 natural log ln of both sides.)1)(1(lnln 2 xxxy
Step 2 Expand the complicated side.
)1ln()1ln(lnln
)1)(1(lnln2
2
xxxy
xxxy
Step 3 Differentiate both side (implicitly for ln y )
1
2
1
11
)1ln()1ln(lnln
2
2
x
x
xxy
y
xxxy
1
2
1
112
x
x
xxy
y
Step 4: Solve for y ’.
1
2
1
112x
x
xxyy
)1)(1( 2 xxxy
Step 5: Substitute y in the above equation and simplify.
1
)1)(1(2
1
)1)(1()1)(1(
1
2
1
11)1)(1(
2
222
22
x
xxxx
x
xxx
x
xxxy
x
x
xxxxxy
1234
221
)1(2)1()1)(1(
23
23323
22
xxx
xxxxxxx
xxxxxxx
Continue to simplify…
1
)1)(1(2
1
)1)(1()1)(1(2
222
x
xxxx
x
xxx
x
xxxy
Let’s double check to make sure that derivative is correct byMultiplying out the original and then taking the derivative.
1234
)1(
)1)(1(
23
23422
2
xxxy
xxxxxxxy
xxxy
Remember this problem was to practice the technique. You would not use it on something this simple.
Example 7 Consider the function y = xx.
Not a power function nor an exponential function.
This is the graph: domain x > 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.
To find the derivative of y = xx , we will take the ln of both sides first and then expand.
xxy
xy x
lnln
lnln
Now, to find the derivative we differentiate both sides implicitly.
xxxyy
xy
y
xxx
y
y
xxy
x ln1ln1
ln1
1ln1
lnln
1
ln1
ln10
ln10
ln1ln1
ex
x
x
xx
xxxyyx
x
To find the critical numbers, set y ’ = 0 and solve for x.
...367.011
eex
Thus, the minimum point occurs at x = 1/e or about 0.367
x
y
Now test x = 0.1 in y ’, y ’(0.1) = -1.034 < 0 and x = 0.5 in y ’, y ’(0.5) = 0.216 > 0
Summary
We learned two rules for differentiating logarithmic functions:
Rule 1: Derivative of lnx
xx
dx
d 1ln 0x
Rule 2: The Chain Rule for Log Functions
)(
)()(ln
xf
xfxf
dx
d 0)( xf
We also learned it can be beneficial to expand a logarithm before you take the derivative and that sometimes it is useful to take thenatural log (ln) of both sides of an equation, rewrite and then takethe derivative implicitly.
Homework
Section 5.1 Pg. 329 7-33 odds, 45-59 odds, 71, 75, 77, 79, 83, 85