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

2. Think of a “derivative” as the slope of the tangent line At a given point of a function. Other notation: (The derivative line is the slope of the line).

3. 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.

4. But which tangent line for this function?

6. Consider a secant line:

7. Definition of the derivative

8. Using the definition to find the derivative- Substitute x+h into the formula: (because h goes to 0!)

9. Using the definition to find the derivative- Substitute f(x) into the formula: Find

10. Using the definition to find the derivative- Substitute f(x) into the formula: Find

11. Do Now: find the derivative of the function Using the definition:

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

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

15. examples: constant multiple rule: 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.

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

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

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

19. Find the first derivative of the following:

20. Can you guess the second derivative?

21. Find the first derivative of the following: the second derivative is:

22. Now: find the derivative of the function Using the power rule: Use exponents first!

23. Now: find the derivative of the function Using the power rule: Put back in fraction form!

24. Applied to trickier questions: Put in exponential form first!

25. Applied to trickier questions: apply the rule:

26. Don’t forget to reduce the exponent by 1!

27. 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.)

33. First derivative (slope) is zero at:

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

35. quotient rule: or

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