1. Chapter 3 Limits and the Derivative
Section 5
Basic Differentiation Properties
(Part 1)

2. Objectives for Section 3.5 Power Rule and Differentiation Properties
The student will be able to:
Calculate the derivative of a constant function.
Apply the power rule.
Apply the constant multiple and sum and difference properties.
Barnett/Ziegler/Byleen Business Calculus 12e

3. The Derivative Recall from the previous lesson:
f(x) is a “slope machine”. 𝑓 ′ (𝑎) will tell you the slope of the line tangent to the graph of f(x) at x=a (if it exists).
f(x) is also the instantaneous rate of change of f(x).
f(x) is also the instantaneous velocity of an object.
In this lesson, you will learn an easier way to find 𝑓 ′ 𝑥 .
Barnett/Ziegler/Byleen Business Calculus 12e

4. Derivative Notation
In the preceding section we defined the derivative of a function. There are several widely used symbols to represent the derivative. Given y = f (x), the derivative of f at x may be represented by any of the following:
f (x) y
𝑑𝑦 𝑑𝑥
Later on, you will see how each of these symbols has its particular advantage in certain situations.
Barnett/Ziegler/Byleen Business Calculus 12e

5. Derivative Rules
In the next several slides, you will learn rules for finding derivatives of:
Constant functions
Power functions
Functions multiplied by constants
These rules will enable you to find 𝑓′(𝑥) easier compared to using a limit.
Note that you will be tested on both methods.
Barnett/Ziegler/Byleen Business Calculus 12e

6. Derivative of a Constant
What is the slope of a constant function?
The graph of f (x) = C is a horizontal line with slope 0, so we would expect f (x) = 0.
Theorem 1. Let y = f (x) = C be a constant function, then
y = f (x) = 0.
Barnett/Ziegler/Byleen Business Calculus 12e

7. Example 1 Derivatives of constant functions: If f(x) = 8 f (x)=0
If y = -4
y = 0
If y = 
𝑑𝑦 𝑑𝑥 =0
𝑑 𝑑𝑥 12=0
Barnett/Ziegler/Byleen Business Calculus 12e

8. Derivative of a Power Function
A function of the form f (x) = xn is called a power function.
(n is a real number)
Theorem 2. (Power Rule) Let y = xn be a power function, then
f (x) = nxn – 1.
THEOREM 2 IS VERY IMPORTANT. IT WILL BE USED A LOT!
Barnett/Ziegler/Byleen Business Calculus 12e

9. Example 2 Derivatives of power functions: If 𝑓 𝑥 = 𝑥 5
If y=𝑥
If 𝑓 𝑥 = 𝑥 5 4
If 𝑦= 3 𝑥
If 𝑓 𝑥 = 1 𝑥 2
then 𝑓 ′ 𝑥 =5 𝑥 5−1 =5 𝑥 4
then 𝑑𝑦 𝑑𝑥 = 1𝑥 1−1 = 𝑥 0 =1
then 𝑓 ′ 𝑥 = 5 4 𝑥 1 4
then 𝑦 ′ = 1 3 𝑥 −2 3
𝑦= 𝑥 1 3
then 𝑓 ′ (𝑥)= −2𝑥 −3
𝑓(𝑥)= 𝑥 −2
Barnett/Ziegler/Byleen Business Calculus 12e

10. Constant Multiple Property
Theorem 3. Let y = k u(x) where k is a constant.
Then y = k  u (x)
In words: The derivative of a constant times a function is the constant times the derivative of the function.
Barnett/Ziegler/Byleen Business Calculus 12e

11. Example 3 Differentiate each function: f (x) = 7x4 𝑦=−3 𝑥 2
𝑓 𝑥 =6 𝑥 2 3
𝑦= 2 3 𝑥 6
𝑓′(𝑥)=28 𝑥 3
𝑦 ′ =−6𝑥
𝑓 ′ 𝑥 =4 𝑥 − 1 3
= 2 𝑥 −6 3
𝑑𝑦 𝑑𝑥 =−4 𝑥 −7
Barnett/Ziegler/Byleen Business Calculus 12e

12. Sum and Difference Properties
Theorem 5.
If y = f (x) = u(x) + v(x),
then y = f (x) = u(x) + v(x).
(this is also true for subtraction)
The derivative of the sum of two differentiable functions is the sum of the derivatives.
The derivative of the difference of two differentiable functions is the difference of the derivatives.
Barnett/Ziegler/Byleen Business Calculus 12e

13. Example 5 Differentiate f (x) = 3x5 + x4 – 2x3 + 5x2 – 7x + 4.
Answer: f (x) = 15x4 + 4x3 – 6x2 + 10x – 7
Find 𝑑𝑦 𝑑𝑥 for 𝑦= 𝑥 −5x
Answer: 𝑑𝑦 𝑑𝑥 = 1 2 𝑥 −1 2 −5
Barnett/Ziegler/Byleen Business Calculus 12e

14. Homework #3-5A: Pg. 185 (25-41 odd, 49) Mammoth opens on Nov. 7th!
Barnett/Ziegler/Byleen Business Calculus 12e

15. Chapter 3 Limits and the Derivative
Section 5
Basic Differentiation Properties
(Part 2)

16. Objectives for Section 3.5 Power Rule and Differentiation Properties
The student will be able to:
Solve applications.
Barnett/Ziegler/Byleen Business Calculus 12e

17. Applications Remember that the derivative has these meanings:
Slope of the tangent line at a point on the graph of a function.
Instantaneous velocity.
Instantaneous rate of change.
Barnett/Ziegler/Byleen Business Calculus 12e

18. Tangent Line Example Let f (x) = x4 – 6x2 + 10. (a) Find f (x)
(b) Find the equation of the tangent line at x = 1
(c) Find the values of x where the tangent line is horizontal.
Solution:
f (x) = 4x3 - 12x
Barnett/Ziegler/Byleen Business Calculus 12e

19. Example (continued) f (x) = x4 – 6x2 + 10.
(b) Find the equation of the tangent line at x = 1
Solution:
Slope: f (1) = 4(13) – 12(1) = -8. Point: If x = 1, then y = f (1) = 5
Point-slope form: y – y1 = m(x – x1) y – 5 = –8(x –1) y = –8x + 13
Barnett/Ziegler/Byleen Business Calculus 12e

20. Barnett/Ziegler/Byleen Business Calculus 12e

21. Example (continued) Let f (x) = x4 – 6x2 + 10.
c) Find the values of x where the tangent line is horizontal.
Solution:
Tangent line is horizontal means slope is zero.
So set derivative = 0 and solve for x.
4𝑥 3 −12𝑥=0
4𝑥 𝑥 2 −3 =0
𝑥=0, ± 3
Barnett/Ziegler/Byleen Business Calculus 12e

22. Barnett/Ziegler/Byleen Business Calculus 12e

23. Instantaneous Velocity
An object moves along the y-axis (marked in feet) so that its position at time x (in seconds) is:
𝑓 𝑥 = 𝑥 3 −15 𝑥 2 +72𝑥
Find the instantaneous velocity function.
Find the velocity at 2 and 5 seconds.
Find the time(s) when the velocity is 0.
Barnett/Ziegler/Byleen Business Calculus 12e

24. Instantaneous Velocity
An object moves along the y-axis (marked in feet) so that its position at time x (in seconds) is:
𝑓 𝑥 = 𝑥 3 −15 𝑥 2 +72𝑥
Find the instantaneous velocity function.
𝑓 ′ 𝑥 =3 𝑥 2 −30𝑥+72
Barnett/Ziegler/Byleen Business Calculus 12e

25. Instantaneous Velocity
An object moves along the y-axis (marked in feet) so that its position at time x (in seconds) is:
𝑓 𝑥 = 𝑥 3 −15 𝑥 2 +72𝑥
Find the instantaneous velocity function.
Find the velocity at 2 and 5 seconds.
𝑓 ′ 𝑥 =3 𝑥 2 −30𝑥+72
𝑓 ′ 2 =24
𝑓 ′ 5 =−3
The velocity at 2 seconds is 24 ft/sec.
The velocity at 5 seconds is −3 ft/sec.
Barnett/Ziegler/Byleen Business Calculus 12e

26. Instantaneous Velocity
An object moves along the y-axis (marked in feet) so that its position at time x (in seconds) is:
𝑓 𝑥 = 𝑥 3 −15 𝑥 2 +72𝑥
Find the time(s) when the velocity is 0.
𝑓 ′ 𝑥 =3 𝑥 2 −30𝑥+72
0=3 𝑥 2 −30𝑥+72
0=3( 𝑥 2 −10𝑥+24)
0=3(𝑥−6)(𝑥−4)
𝑥=4, 6
𝑇ℎ𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑖𝑠 0 𝑎𝑡 4 𝑎𝑛𝑑 6 𝑠𝑒𝑐𝑜𝑛𝑑𝑠.
Barnett/Ziegler/Byleen Business Calculus 12e

27. Instantaneous Rate of Change
If C(x) is the total cost of producing x items, then C(x) is the instantaneous rate of change of cost at a production level of x items.
Barnett/Ziegler/Byleen Business Calculus 12e

28. Application Example
The total cost (in dollars) of producing x portable radios per day is
C(x) = x – 0.5x2
for 0 ≤ x ≤ 100.
Find 𝐶 ′ 𝑥
Solution: C(x) = 100 – x.
Barnett/Ziegler/Byleen Business Calculus 12e

29. Example (continued) Find C(80) and 𝐶′(80) and interpret these results.
Solution:
𝐶 80 = (80) – 0.5(80)2=5800
C(80) = 100 – 80 = 20
At a production level of 80 radios, the total cost is $5800 and is increasing at a rate of $20 per radio.
Barnett/Ziegler/Byleen Business Calculus 12e

30. #3-5B: Pg. 185 (32-42 even, 50, 55, 56, 81, 87, 89) Homework
Barnett/Ziegler/Byleen Business Calculus 12e