Differentiation Rules and Horizontal Tangents: A Comprehensive Guide

1. Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington 3.1-3.2 Differentiation Rules Colorado National Monument

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

3. We saw that if , . power rule This is part of a pattern. examples:

4. constant multiple rule: examples:

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

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

12. First derivative (slope) is zero at:

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

14. quotient rule: or

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