3.3 Rules for Differentiation

1. Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington 3.3 Rules for Differentiation 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. (Pascal’s Triangle) If we find derivatives with the difference quotient: We observe a pattern: …

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

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. 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.)

13. First derivative (slope) is zero at:

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

15. quotient rule: or

16. 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