1. 3.3 Rules for Differentiation
Colorado National Monument
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2003

2. If the derivative of a function is its slope, then for a constant function, the derivative must be zero.
example:
The derivative of a constant is zero.

3. If we find derivatives with the difference quotient:
(Pascal’s Triangle)
We observe a pattern: …

4. We observe a pattern: …
examples:
power rule

5. constant multiple rule:
examples:
When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.

6. constant multiple rule:
sum and difference rules:
(Each term is treated separately)

7. Let’s look back at today’s bell-ringer.
Use the rules to find the derivative:

8. Find for the following functions 1. 2. 3. 4. 5. 6.

9. Homework:
Page #1-5 all parts

10. Bell-ringer 10/7 Find the derivative of the following functions: 1.
(hint: split up the numerator)

11. Example:
Find the horizontal tangents of:
Horizontal tangents occur when slope = zero.
Plugging the x values into the original equation, we get:
(The function is even, so we only get two horizontal tangents.)

17. First derivative (slope) is zero at:

18. Find the horizontal tangents:
Ex:
Step 1: Find
Step 2: Set equal to zero.
Step 3: Solve for x.

19. product rule:
Notice that this is not just the product of two derivatives.
This is sometimes memorized as:

20. Use the product rule to find1. 2.

21. quotient rule: or

22. Use the quotient rule to find1. 2.

23. Homework
Pg 37 #6, 7 all and horizontal tangent questions a, b, and c on page 38.

24. Higher Order Derivatives:
is the first derivative of y with respect to x.
is the second derivative.
(y double prime)
is the third derivative.
We will learn later what these higher order derivatives are used for.
is the fourth derivative. p

25. Find the following:
Find if
Find if

26. 1. Solution

27. 2. Solution