Section 3 - Tredyffrin/Easttown School District
3.1 Techniques of Differentiation
Learning Objectives
A student will be able to:
• Use various techniques of differentiations to find the derivatives of various functions.
• Compute derivatives of higher orders.
Up to now, we have been calculating derivatives by using the definition. In this section, we will develop formulas and theorems that will calculate derivatives in more efficient and quick ways. It is highly recommended that you become very familiar with all of these techniques.
The Derivative of a Constant
If [pic] where [pic] is a constant, then [pic].
In other words, the derivative or slope of any constant function is zero.
Proof:
[pic]
Example 1:
If [pic] for all [pic], then [pic] for all [pic]. We can also write [pic].
The Power Rule
If [pic]is a positive integer, then for all real values of [pic], [pic]. The proof of the power rule is omitted in this text, but it is available at and also in video form at Khan Academy Proof of the Power Rule. Note that this proof depends on using the binomial theorem from Precalculus.
[pic].
Example 2:
If [pic], then
[pic]
and
[pic]
The Power Rule and a Constant
If [pic]is a constant and [pic]is differentiable at all [pic], then
[pic]
In simpler notation,
[pic]
In other words, the derivative of a constant times a function is equal to the constant times the derivative of the function.
Example 3:
[pic]
Example 4:
[pic]
Derivatives of Sums and Differences
If [pic]and [pic]are two differentiable functions at [pic], then
[pic]
and
[pic]
In simpler notation,
[pic]
[pic]
The Product Rule
If [pic]and [pic]are differentiable at [pic], then
[pic]
In a simpler notation,
[pic]
The derivative of the product of two functions is equal to the first times the derivative of the second plus the second times the derivative of the first.
Keep in mind that
[pic]
Example 7:
Find [pic]for [pic]
Solution:
There are two methods to solve this problem. One is to multiply the product and then use the derivative of the sum rule. The second is to directly use the product rule. Either rule will produce the same answer. We begin with the sum rule.
[pic]
Taking the derivative of the sum yields
[pic]
Now we use the product rule,
[pic]
which is the same answer.
The Quotient Rule
If [pic]and [pic]are differentiable functions at [pic]and [pic], then
[pic]
In simpler notation,
[pic]
The derivative of a quotient of two functions is the bottom times the derivative of the top minus the top times the derivative of the bottom all over the bottom squared.
Keep in mind that the order of operations is important (because of the minus sign in the numerator) and
[pic]
Example 8:
Find [pic]for
[pic]
Solution:
[pic]
Example 9:
At which point(s) does the graph of [pic]have a horizontal tangent line?
Solution:
Since the slope of a horizontal line is zero, and since the derivative of a function signifies the slope of the tangent line, then taking the derivative and equating it to zero will enable us to find the points at which the slope of the tangent line equals to zero, i.e., the locations of the horizontal tangents.
[pic]
Multiplying by the denominator and solving for [pic],
[pic]
Therefore the tangent line is horizontal at [pic]
Higher Derivatives
If the derivative [pic]of the function [pic]is differentiable, then the derivative of [pic], denoted by [pic], is called the second derivative of [pic]. We can continue the process of differentiating derivatives and obtain third, fourth, fifth and higher derivatives of [pic]. They are denoted by [pic], [pic], [pic], [pic], [pic]
Example 10:
Find the fifth derivative of [pic].
Solution:
[pic]
Example 11:
Show that [pic]satisfies the differential equation [pic]
Solution:
We need to obtain the first, second, and third derivatives and substitute them into the differential equation.
[pic]
Substituting,
[pic]
which satisfies the equation.
Review Questions
Use the results of this section to find the derivatives [pic].
1. [pic]
2. y = [pic]
3. [pic]
4. [pic] (where a and b are constants)
5. [pic]
6. [pic]
7. [pic]
8. [pic]
9. [pic]
10. [pic]
11. Newton’s Law of Universal Gravitation states that the gravitational force between two masses (say, the earth and the moon), m and M, is equal to their product divided by the square of the distance r between them. Mathematically, [pic] where G is the Universal Gravitational Constant [pic]. If the distance r between the two masses is changing, find a formula for the instantaneous rate of change of F with respect to the separation distance r.
12. Find [pic], where [pic] is a constant.
13. Find[pic], where [pic].
Review Answers
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
8. [pic]
9. [pic]
10. [pic]
11. [pic]
12. [pic]
13. -120
Constant and Power Rule Practice
Use the Constant and Power Rules to find the derivative.
1. [pic] 2. [pic]
3. [pic] 4. [pic]
5. [pic] 6. [pic]
7. [pic] 8. [pic]
9. [pic] 10. [pic]
11. [pic] 12. [pic]
13. [pic] 14. [pic]
15. [pic] 16. [pic]
Find the value of the derivative of the function at the indicated point.
17. [pic] 18. [pic]
Find the equation of the tangent line to the graph of the function at the indicated point
19. [pic] 20. [pic]
Answers:
1. [pic] 2. [pic] 3. [pic] 4. [pic] 5. [pic]
6. [pic] 7. [pic] 8. [pic] 9. [pic] 10. [pic]
11. [pic] 12. [pic] 13. [pic] 14. [pic]
15. [pic] 16. [pic] 17. [pic] 18. [pic]
19. [pic] 20. [pic]
Power Rule HW
Find the derivative of each function. In your answers, rational exponents are OK, negative exponents are not.
1.) [pic]
2.) [pic]
3.) [pic]
4.) [pic] POWER RULE: [pic]
5.) [pic]
6.) [pic]
7.) [pic]
Answers:
1.) [pic]
2.) [pic]
3.) [pic]
4.) [pic]
5.) [pic]
6.) [pic]
7.) [pic]
Mo’ Power Rule HW
Find the derivative of each function. Make sure your answers are factored completely. If a point is given, find the value of the derivative at that point.
1.) [pic] Point: [pic]
2.) [pic] Point: (0, 1)
3.) [pic] Point: (1, –3)
4.) [pic]
5.) [pic] Point: (7, 350)
6.) [pic]
7.) [pic] Point: (–2, –512)
8.) [pic]
Answers:
1.) [pic]
2.) [pic]
3.) [pic]
4.) [pic]
5.) [pic]
6.) [pic]
7.) [pic]
8.) [pic]
Product Rule Practice
Find the derivative using the Product Rule. Final answer should be in simplest form.
1. [pic]
2. [pic]
3. [pic]
4. [pic]
Determine the point(s), if any, at which the graph of the function has a horizontal tangent line.
5. [pic] 6. [pic]
Find the derivative. Do not use the Product Rule.
7. [pic] 8. [pic]
Answers:
1. [pic] 2. [pic]
3. [pic] 4. [pic]
5. [pic] 6. none 7. [pic]
8. [pic]
Product Rule HW
Find the derivative of each function. Make sure your answers are factored completely. If a point is given, find the value of the derivative at that point.
1.) [pic] Point: (1, 2)
2.) [pic] Point: [pic]
3.) [pic] Point: (4, 6)
4.) [pic]
5.) [pic] PRODUCT RULE: [pic]
6.) [pic] Point: (2, 36)
7.) [pic]
8.) [pic]
Answers:
1.) [pic]
2.) [pic]
3.) [pic]
4.) [pic]
5.) [pic]
6.) [pic]
7.) [pic]
8.) [pic]
Quotient Rule Practice
Use the Quotient Rule to find the derivative. Final answers should be in simplest form.
1. [pic]
2. [pic]
3. [pic]
Find the equation of the line tangent to [pic] at the indicated point.
4. [pic] Point: (6, 6)
5. [pic] Point: [pic]
Find the derivative without the use of the Product or Quotient Rules. Give simplified final answers.
6. [pic]
7. [pic]
Answers:
1. [pic] 2. [pic] 3. [pic]
4. [pic] 5. [pic] 6. [pic]
7. [pic]
Quotient Rule HW
Find the derivative of each function. Make sure your answers are factored completely. If a point is given, find the value of the derivative at that point.
1.) [pic] Point: (6, 6)
2.) [pic] Point: [pic]
3.) [pic] QUOTIENT RULE: [pic]
4.) [pic]
5.) [pic] (Do not use the product or quotient rules.)
6.) [pic] (Do not use the quotient rule.)
7.) [pic] Point: (2, 1)
8.) [pic]
Answers:
|1.) [pic] |5.) [pic] |
| | |
|2.) [pic] |6.) [pic] |
| | |
|3.) [pic] |7.) [pic] |
| | |
|4.) [pic] |8.) [pic] |
Practice Problems on 3.1 (Constant, Power, Product & Quotient Rules)
Differentiate. Remember to simplify the function to make differentiating easier. Final answers should be in simplest form.
1. [pic] 2. [pic]
3. [pic] 4. [pic]
5. [pic] 6. [pic]
7. [pic] 8. [pic]
9. [pic] 10. [pic]
11. [pic] 12. [pic]
Find the equation of the line tangent to the function at the indicated [pic] value.
13. [pic] [pic]
Find the slope of the graph at the indicated point.
14. [pic] [pic]
Find the point(s), if any, at which the graph of the function has a horizontal tangent.
15. [pic]
Answers:
1. [pic] 2. [pic] 3. [pic] 4. [pic]
5. [pic] 6. [pic] 7. [pic] 8. [pic]
9. [pic] 10. [pic] 11. [pic]
12. [pic] 13. [pic] 14. [pic]
15. [pic]
3.2 Derivatives of Trigonometric Functions
Learning Objectives
A student will be able to:
• Compute the derivatives of various trigonometric functions.
Recall from Chapter 1 that if the angle [pic]is measured in radians,
[pic]and [pic]
We now want to find an expression for the derivative of the six trigonometric functions [pic]and [pic]. We first consider the problem of differentiating [pic], using the definition of the derivative.
[pic]
Since
[pic]
The derivative becomes
[pic]
Therefore,
[pic]
It will be left as an exercise to prove that
[pic]
The derivatives of the remaining trigonometric functions follow.
Derivatives of Trigonometric Functions
[pic]
Keep in mind that for all the derivative formulas for the trigonometric functions, the argument [pic]is measured in radians.
Example 1:
Show that [pic]
Solution:
It is possible to prove this relation by the definition of the derivative. However, we use a simpler method.
Since
[pic]
then
[pic]
Example 2:
Find [pic].
Solution:
Using the product rule and the formulas above, we obtain
[pic]
Example 3:
Find [pic]if [pic]. What is the slope of the tangent line at [pic]?
Solution:
Using the quotient rule and the formulas above, we obtain
[pic]
To calculate the slope of the tangent line, we simply substitute [pic]:
[pic]
We finally get the slope to be approximately
[pic]
Example 4:
If [pic], find [pic].
Solution:
[pic]
Substituting for [pic],
[pic]
Thus [pic].
Multimedia Links
For examples of finding the derivatives of trigonometric functions (4.4), see Math Video Tutorials by James Sousa, The Derivative of Sine and Cosine (9:20)[pic].
Review Questions
Find the derivative [pic] of the following functions:
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
8. [pic]
9. If [pic], find [pic]
10. Use the definition of the derivative to prove that [pic]
Review Answers
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
8. [pic]
9. [pic]
Practice on Derivatives involving Trig Functions
Find [pic] for each of the following. Use trig identities as appropriate to help you simplify.
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
8. [pic]
9. [pic]
10. Find the derivative of the function [pic] at the point [pic].
Answers:
1. [pic]
2. [pic] (Did you use the product rule? You didn’t need to.)
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic] (Did you use the quotient rule? You didn’t need to.)
8. [pic]
9. [pic]
10. [pic]
3.3 The Chain Rule
Learning Objectives
A student will be able to:
• Know the chain rule and its proof.
• Apply the chain rule to the calculation of the derivative of a variety of composite functions.
We want to derive a rule for the derivative of a composite function of the form [pic]in terms of the derivatives of f and g. This rule allows us to differentiate complicated functions in terms of known derivatives of simpler functions.
The Chain Rule
If [pic]is a differentiable function at [pic]and [pic]is differentiable at [pic], then the composition function [pic]is differentiable at [pic]. The derivative of the composite function is:
[pic]
Another way of expressing, if [pic]and [pic], then
[pic]
And a final way of expressing the chain rule is the easiest form to remember: If [pic]is a function of [pic]and [pic]is a function of [pic], then
[pic]
Example 1:
Differentiate [pic]
Solution:
Using the chain rule, let [pic]Then
[pic]
The example above is one of the most common types of composite functions. It is a power function of the type
[pic]
The rule for differentiating such functions is called the General Power Rule. It is a special case of the Chain Rule.
The General Power Rule
if
[pic]
then
[pic]
In simpler form, if
[pic]
then
[pic]
Example 2:
What is the slope of the tangent line to the function [pic]that passes through point [pic]?
Solution:
We can write [pic]This example illustrates the point that [pic]can be any real number including fractions. Using the General Power Rule,
[pic]
To find the slope of the tangent line, we simply substitute [pic]into the derivative:
[pic]
Example 3:
Find [pic]for [pic].
Solution:
The function can be written as [pic]Thus
[pic]
Example 4:
Find [pic]for [pic]
Solution:
Let [pic]By the chain rule,
[pic]
where [pic]Thus
[pic]
Example 5:
Find [pic]for [pic]
Solution:
This example applies the chain rule twice because there are several functions embedded within each other.
Let [pic]be the inner function and [pic]be the innermost function.
[pic]
Using the chain rule,
[pic]
Notice that we used the General Power Rule and, in the last step, we took the derivative of the argument.
Multimedia Links
For an introduction to the Chain Rule (5.0), see Khan Academy, Calculus: Derivatives 4: The Chain Rule (9:11)[pic].
For more examples of the Chain Rule (5.0), see [ Math Video Tutorials by James Sousa The Chain Rule: Part 1 of 2 (8:45)[pic]. and [ Math Video Tutorials by James Sousa The Chain Rule: Part 2 of 2 (8:35)[pic].
Review Questions
Find [pic].
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
8. [pic]
9. [pic]
10. [pic]
11. [pic]
Review Answers
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
8. [pic] or [pic]
9. [pic]
10. [pic]
which simplifies to [pic]
11. [pic]
Chain Rule Practice
Differentiate. Simplify answers.
1. [pic] 2. [pic]
3. [pic] 4. [pic]
5. [pic] 6. [pic]
7. [pic] 8. [pic]
9. [pic] 10. [pic]
11. Find the equation of the tangent line to the graph of [pic] at the point [pic].
12. Find the slope of [pic] when [pic].
Answers:
1. [pic] 2. [pic]
3. [pic] 4. [pic]
5. [pic] 6.[pic]
7. [pic] 8. [pic]
9. [pic] 10. [pic]
11. [pic] 12. [pic]
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