section 1 - radford · web viewsection 2.4: the chain rule practice hw from larson textbook (not to...
TRANSCRIPT
![Page 1: Section 1 - Radford · Web viewSection 2.4: The Chain Rule Practice HW from Larson Textbook (not to hand in) p. 122 # 1-23 odd, 33-97 odd In this section, we want to differentiate](https://reader036.vdocument.in/reader036/viewer/2022071402/60ef4da653c03d22e86933e2/html5/thumbnails/1.jpg)
Section 2.4: The Chain Rule
Practice HW from Larson Textbook (not to hand in)p. 122 # 1-23 odd, 33-97 odd
In this section, we want to differentiate different type of function compositions. The rules displayed are all special cases of the chain rule, which we discuss later in the section.
General Power Rule
Example 1: Differentiate .
Solution:
█
Example 2: Differentiate .
Solution:
█
1
![Page 2: Section 1 - Radford · Web viewSection 2.4: The Chain Rule Practice HW from Larson Textbook (not to hand in) p. 122 # 1-23 odd, 33-97 odd In this section, we want to differentiate](https://reader036.vdocument.in/reader036/viewer/2022071402/60ef4da653c03d22e86933e2/html5/thumbnails/2.jpg)
Example 3: Differentiate .
Solution: Using the exponent law , start by rewriting the function as
. Then using the general power rule, we calculate the derivative as follows.
█
Example 4: Differentiate .
Solution:
█
Derivative of an Exponential Function for an Arbitrary Base
2
![Page 3: Section 1 - Radford · Web viewSection 2.4: The Chain Rule Practice HW from Larson Textbook (not to hand in) p. 122 # 1-23 odd, 33-97 odd In this section, we want to differentiate](https://reader036.vdocument.in/reader036/viewer/2022071402/60ef4da653c03d22e86933e2/html5/thumbnails/3.jpg)
Base e:
Arbitrary Base:
Example 5: Differentiate .
Solution:
█
Chain Rule For Exponential Functions
Base e:
Arbitrary Base:
Example 6: Differentiate .
Solution:
█
3
![Page 4: Section 1 - Radford · Web viewSection 2.4: The Chain Rule Practice HW from Larson Textbook (not to hand in) p. 122 # 1-23 odd, 33-97 odd In this section, we want to differentiate](https://reader036.vdocument.in/reader036/viewer/2022071402/60ef4da653c03d22e86933e2/html5/thumbnails/4.jpg)
Example 7: Differentiate .
Solution:
█
Example 8: Differentiate .
4
![Page 5: Section 1 - Radford · Web viewSection 2.4: The Chain Rule Practice HW from Larson Textbook (not to hand in) p. 122 # 1-23 odd, 33-97 odd In this section, we want to differentiate](https://reader036.vdocument.in/reader036/viewer/2022071402/60ef4da653c03d22e86933e2/html5/thumbnails/5.jpg)
Solution: Since this function is a product of two functions of t, we need to use the product rule. When taking the derivative of the second function, we apply the chain rule applied to the exponential function of base e. This gives
█
Chain Rule
Used to take the derivative of a composition of two functions.
Chain Rule
Note: The general power rule and chain rule for the exponential function are special cases of the Chain Rule.
5
![Page 6: Section 1 - Radford · Web viewSection 2.4: The Chain Rule Practice HW from Larson Textbook (not to hand in) p. 122 # 1-23 odd, 33-97 odd In this section, we want to differentiate](https://reader036.vdocument.in/reader036/viewer/2022071402/60ef4da653c03d22e86933e2/html5/thumbnails/6.jpg)
Example 9: Differentiate using the general power rule and the chain rule.
Solution:
Chain Rule Applied to the Trigonometric Functions
6
![Page 7: Section 1 - Radford · Web viewSection 2.4: The Chain Rule Practice HW from Larson Textbook (not to hand in) p. 122 # 1-23 odd, 33-97 odd In this section, we want to differentiate](https://reader036.vdocument.in/reader036/viewer/2022071402/60ef4da653c03d22e86933e2/html5/thumbnails/7.jpg)
Example 10: Differentiate .
Solution:
█
Example 11: Differentiate .
Solution:
█
Example 12: Differentiate .
Solution:
Example 13: Differentiate .
7
![Page 8: Section 1 - Radford · Web viewSection 2.4: The Chain Rule Practice HW from Larson Textbook (not to hand in) p. 122 # 1-23 odd, 33-97 odd In this section, we want to differentiate](https://reader036.vdocument.in/reader036/viewer/2022071402/60ef4da653c03d22e86933e2/html5/thumbnails/8.jpg)
Solution:
Formulas for the Derivative of Logarithmic Functions
Natural Logarithm:
Logarithm of an Arbitrary Base:
8
![Page 9: Section 1 - Radford · Web viewSection 2.4: The Chain Rule Practice HW from Larson Textbook (not to hand in) p. 122 # 1-23 odd, 33-97 odd In this section, we want to differentiate](https://reader036.vdocument.in/reader036/viewer/2022071402/60ef4da653c03d22e86933e2/html5/thumbnails/9.jpg)
Example 14: Differentiate
Solution:
█
Chain Rule for the Logarithmic Functions
Natural Logarithm:
Logarithm of an Arbitrary Base:
9
![Page 10: Section 1 - Radford · Web viewSection 2.4: The Chain Rule Practice HW from Larson Textbook (not to hand in) p. 122 # 1-23 odd, 33-97 odd In this section, we want to differentiate](https://reader036.vdocument.in/reader036/viewer/2022071402/60ef4da653c03d22e86933e2/html5/thumbnails/10.jpg)
Example 15: Differentiate .
Solution:
█
Example 16: Differentiate .
Solution:
█
10
![Page 11: Section 1 - Radford · Web viewSection 2.4: The Chain Rule Practice HW from Larson Textbook (not to hand in) p. 122 # 1-23 odd, 33-97 odd In this section, we want to differentiate](https://reader036.vdocument.in/reader036/viewer/2022071402/60ef4da653c03d22e86933e2/html5/thumbnails/11.jpg)
Example 17: Differentiate .
Solution: Using the chain rule formula for the natural logarithm of an absolute value
, we calculate the derivative as follows:
Example 18: Differentiate .
Solution: On this one, it is easier to use some properties of logarithms to rewrite the function before differentiating. We do this as follows:
We now apply the chain rule for the ln function to differentiate as follows:
█
11