1. The Constant and Power Rules for Derivatives
Calculus 1
Ms. Penabad
2014

2. In today’s lesson we will cover:
Theorem 3.2: Constant Rule
Example 1
Example 2
Example 3
Theorem 3.3: Power Rule

3. Constant Rule
Theorem 3.2: Constant Rule
If c is a real number, then

4. Let’s evaluate the following derivative:
Example 1
Let’s evaluate the following derivative:

5. Let’s evaluate the following derivative:
Example 1
Let’s evaluate the following derivative:
RECALL: Constant Rule

6. Let’s evaluate the following derivative:
Example 1
RECALL: Constant Rule
Let’s evaluate the following derivative:
Since 2014 is a real number, it is considered to be a constant. By the Constant Rule, the derivative of ANY constant will always equal 0.

7. Let’s evaluate the following derivative:
Example 2
Let’s evaluate the following derivative:

8. Let’s evaluate the following derivative:
Example 2
Let’s evaluate the following derivative:
RECALL: Constant Rule

9. Let’s evaluate the following derivative:
Example 2
RECALL: Constant Rule
Let’s evaluate the following derivative:
Step 1: if visualization needed: simplify exponent (-8) by moving e to the denominator.

10. Let’s evaluate the following derivative:
Example 2
RECALL: Constant Rule
Let’s evaluate the following derivative:
Step 2: since
is a constant, by the
Constant Rule,
Step 1: if visualization needed: simplify exponent (-8) by moving e to the denominator.

11. Let’s evaluate the following derivative:
Example 3
Let’s evaluate the following derivative:

12. Let’s evaluate the following derivative:
Example 3
Let’s evaluate the following derivative:
RECALL: Constant Rule

13. Example 3 Let’s evaluate the following derivative:
RECALL: Constant Rule
Let’s evaluate the following derivative:
Step 1: if visualization needed: simplify fraction.

14. Example 3 Let’s evaluate the following derivative:
RECALL: Constant Rule
Let’s evaluate the following derivative:
Step 2: since 2.6 is a constant, by the Constant Rule, the answer is 0.
Step 1: if visualization needed: simplify fraction.

15. Power Rule
Theorem 3.2: Power Rule
If n is a positive integer, then

16. Let’s evaluate the following derivative:
Example 1
Let’s evaluate the following derivative:

17. Let’s evaluate the following derivative:
Example 1
Let’s evaluate the following derivative:
RECALL: Power Rule

18. Let’s evaluate the following derivative:
Example 1
RECALL: Power Rule
Let’s evaluate the following derivative:

19. Example 1 Let’s evaluate the following derivative: RECALL: Power Rule
Step 1: notice x has a coefficient of 1

20. Example 1 Let’s evaluate the following derivative: RECALL: Power Rule
Step 1: notice x has a coefficient of 1
Step 2: move the 9 out front
Step 3: subtract 1 from the exponent

21. Example 1 Let’s evaluate the following derivative: RECALL: Power Rule
Step 4: simplify
coefficient (1·9)=9
exponent (9-1)=8
Step 1: notice x has a coefficient of 1
Step 2: move the 9 out front
Step 3: subtract 1 from the exponent

22. Let’s evaluate the following derivative:
Example 2
Let’s evaluate the following derivative:

23. Let’s evaluate the following derivative:
Example 2
Let’s evaluate the following derivative:
RECALL: Power Rule

24. Let’s evaluate the following derivative:
Example 2
RECALL: Power Rule
Let’s evaluate the following derivative:

25. Example 2 Let’s evaluate the following derivative: RECALL: Power Rule
Step 1: notice x has an exponent of 1

26. Example 2 Let’s evaluate the following derivative: RECALL: Power Rule
Step 1: notice x has an exponent of 1
Step 2: move the 1 out front
Step 3: subtract 1 from the exponent

27. Example 2 Let’s evaluate the following derivative: RECALL: Power Rule
Step 4: simplify (1-1)=0
Step 1: notice x has an exponent of 1
Step 2: move the 1 out front
Step 3: subtract 1 from the exponent

28. Example 2 Let’s evaluate the following derivative: RECALL: Power Rule
Step 4: simplify (1-1)=0
Step 5: simplify =1
Step 1: notice x has an exponent of 1
Step 2: move the 1 out front
Step 3: subtract 1 from the exponent

29. Let’s evaluate the following derivative:
Example 3
Let’s evaluate the following derivative:

30. Let’s evaluate the following derivative:
Example 3
Let’s evaluate the following derivative:
RECALL: Power Rule

31. Example 3 Let’s evaluate the following derivative: RECALL: Power Rule
Step 1: move the 8 out front
Step 2: subtract 1 from the exponent

32. Example 3 Let’s evaluate the following derivative: RECALL: Power Rule
Step 3: simplify
coefficient (2·8)=16
exponent (8-1)=7
Step 1: move the 8 out front
Step 2: subtract 1 from the exponent

33. Now…LET’S PRACTICE