德行高尚的人为人处事知深浅明尊卑

y’ = x + 2xlnx
This can also be writBaidu Nhomakorabeaen y’ = x(1+2lnx)
Here is the second rule for differentiating logarithmic functions.
Rule 4: The Chain Rule for Log Functions
x 2 1 x3 2 6
x2 1 x3 2 6
y 2x 9x3 9x x 3 2 2x 10x3 9x 2 20x4 18x2 4x
x2 1 x3 2
x2 1 2 x3 2 7
x2 1 x3 2
y ln x2 1x3 2 6
The inside, x² +1
Example 4: Differentiate y ln x2 1x3 2 6
Solution: There are two ways to do this problem. One is easy and the other is more difficult.
5.5 Differentiation of Logarithmic Functions
By Dr. Julia Arnold and Ms. Karen Overman using Tan’s 5th edition Applied Calculus for the managerial , life, and social sciences text
Try again. Return to previous slide.
Good work! Using the product rule:
F’(x) = (1st)(derivative of 2nd) + (2nd)(derivative of 1st)
y’
=
x²
1 x
+ (lnx)(2x)
The easy way requires that we simplify the log using some of the expansion properties.
Now we will find derivatives of logarithmic functions and we will Need rules for finding their derivatives.
Rule 3: Derivative of ln x
d dx
ln
x
1 x
x 0
Let’s see if we can discover why the rule is as above.
Solution: This derivative will require the product rule.
f(x) xlnx
f (x) x 1 lnx 1 x
Product Rule: (1st)(derivative of 2nd) + (2nd)(derivative of 1st)
First define the natural log function as follows: y ln x
Now rewrite in exponential form: ey x
Now differentiate implicitly:
eyy 1
y
1 ey
1 x
Example 1: Find the derivative of f(x)= xlnx.
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)
d ln f(x) f(x)
dx
f(x)
f(x) 0
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)
f (x) 1 lnx
Example 2: Find the derivative of g(x)= lnx/x
Solution: This derivative will require the quotient rule.
g(x) lnx x
1
x
lnx 1 Quotient Rule:
g(x) x
(bottom)(derivative of top) – (top)(derivative of bottom)
x2
(bottom)²
g(x) 1 lnx x2
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?
Example 3: Find the derivative of f(x) ln(x2 1)
Solution: Using the chain rule for logarithmic functions.
f(x) ln(x2 1)
f(x)
2x x2 1
Derivative of the inside, x² +1
The difficult way:
y
d dx
x2
1
x3
2
6
x 2 1 6 x3 2 5 3x2
x3 2 6 2x
x2
1
x3
2
6
x2
1
x3
2
6
y 18x2 x 2 1 x 3 2 5 2x x 3 2 6 2x x 3 2 5 9x x 2 1 x 3 2