The Power Rule

1. The Power Rule • If we are given a power function: • Then, we can find its derivative using the following shortcut rule, called the POWER RULE:

2. An example • If • Then,

3. Try it on the following examples

4. Answers – (You should be able to fill in the middle steps)

5. Examples with radicals –(You should have reviewed power functions first.)

6. Examples with radicals –(Review negative powers as needed, too.)

7. Examples with fractions.

8. Examples with fractions.

9. The Constant Multiple Rule • If we are given a function multiplied by a constant, as follows: • Then, we can find its derivative as follows • Essentially, we just move the constant multiple into the derivative.

10. An Example Take the derivative using the power rule.

11. Now you try a few: • Find the derivatives:

12. Answers • Find the derivatives:

13. The Derivative of a Constant • Here’s the rule: the derivative of a constant is zero. • Here’s what it means graphically: y = 5 is a horizontal line with a slope of 0 The straight line most like the line is itself, so the derivative is the slope of the same horizontal line. The slope of any horizontal line is 0, so the derivative of any constant function is always 0. Example:

14. The Sum Rule • If we have a function given to us which is a sum of terms, then we can find its derivative by taking the derivative of each of the terms individually. • Example: Derivative of x2 is 2x Derivative of 4x is 4 Derivative of 6 is 0 - +

15. Now you try a few: • Find the derivatives:

16. Answers: Use the power rule on each term. Rewrite f(x) so that each term is written as a power of x.

17. Using the power rule: • Given , determine the slope of the line tangent to the graph at • Then determine an equation of the tangent line at the same value. • Graph both.

18. Using the power rule: • Given , determine the slope of the line tangent to the graph at

19. Using the power rule: • Given , determine the equation of the line tangent to the graph at • Slope is ¼. • To find the point on the graph:

20. Find the equation. • Using the format , substitute for

21. Using the power rule: • Graph both.

22. Other wording . . . • Another way to ask you to take the derivative of a function is to ask you to differentiate the function.

23. Another notation. • Another notation (besides ) for the first derivative is • So, when the text says: “Find the rate of change where ” , and they tell you that it means to find the derivative when x = 4: