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Differentiation
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2. Higher-Order Derivatives
2.6
Higher-Order Derivatives
Copyright © Cengage Learning. All rights reserved.

3. Objectives Find higher-order derivatives.
Find and use a position function to determine the velocity and acceleration of a moving object.

4. Second, Third, and Higher-Order Derivatives

5. Second, Third, and Higher-Order Derivatives
The “standard” derivative f  is often called the first derivative of f. The derivative of f  is the second derivative of f and is denoted by f .
The derivative of f  is the third derivative of f and is denoted by f .

6. Second, Third, and Higher-Order Derivatives
By continuing this process, you obtain higher-order derivatives of f.
Higher-order derivatives are denoted as follows.

7. Example 2 – Finding Higher-Order Derivatives
Find the value of g(2) for the function
Solution:
Begin by differentiating three times.

8. Example 2 – Solution
cont’d
Then, evaluate the third derivative of g at t = 2.

9. Second, Third, and Higher-Order Derivatives
Note that with each successive differentiation, the degree of the polynomial drops by one. Eventually, higher-order derivatives of polynomial functions degenerate to a constant function.

10. Second, Third, and Higher-Order Derivatives
Specifically, the nth-order derivative of an nth-degree polynomial function
is the constant function
where
Each derivative of order higher than n is the zero function.

11. Example 3 – Finding Higher-Order Derivatives
Find the first four derivatives of
Solution:

12. Example 1 – Solution
cont’d

13. Acceleration

14. Acceleration
You have seen that the velocity of a free-falling object (neglecting air resistance) is given by the derivative of its position function.
In other words, the rate of change of the position with respect to time is defined to be the velocity.
In a similar way, the rate of change of the velocity with respect to time is defined to be the acceleration of the object.

15. Acceleration
To find the position, velocity, or acceleration at a particular time t, substitute the given value of t into the appropriate function, as illustrated in Example 4.

16. Example 4 – Finding Acceleration
A ball is thrown upward from the top of a 160-foot cliff, as shown in Figure 2.32.
The initial velocity of the ball is 48 feet per second, which implies that the position function is
where the time t is measured in seconds. Find the height, velocity, and acceleration of the ball at t = 3.
Figure 2.32

17. Example 4 – Solution
Begin by differentiating to find the velocity and acceleration functions.

18. Example 4 – Solution
cont’d
To find the height, velocity, and acceleration at t = 3, substitute t = 3 into each of the functions above.

19. Acceleration
In Example 4, notice that the acceleration of the ball is –32 feet per second squared at any time t. This constant acceleration is due to the gravitational force of Earth and is called the acceleration due to gravity.
Note that the negative value indicates that the ball is being pulled down—toward Earth.

20. Acceleration
The following position function neglects air resistance, which is appropriate because the moon has no atmosphere—and no air resistance.
This means that the position function for any free-falling object on the moon is given by
where s is the height (in feet), t is the time (in seconds), v0 is the initial velocity (in feet per second), and h0 is the initial height (in feet).

21. Acceleration
This position function is valid for all objects, whether heavy ones such as hammers or light ones such as feathers.

22. Example 6 – Finding Velocity and Acceleration
The velocity v (in feet per second) of a certain automobile starting from rest is
where t is the time (in seconds).
The positions of the automobile at 10-second intervals are shown in Figure 2.33.
Find the velocity and acceleration of the automobile at 10-second intervals from t = 0 to t = 60.
Figure 2.33

23. Example 6 – Solution
To find the acceleration function, differentiate the velocity function.

24. Example 6 – Solution
cont’d
In the table, note that the acceleration approaches zero as the velocity levels off.
This observation should agree with your experience—when riding in an accelerating automobile, you do not feel the velocity, but you do feel the acceleration.
In other words, you feel changes in velocity.