2.3 Rules for Differentiation Colorado National Monument Vista High, AB Calculus. Book Larson, V9 2010Photo by Vickie Kelly, 2003

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.

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

examples: power rule We observe a pattern:…

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.

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

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.)

First derivative (slope) is zero at:

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

quotient rule: or

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. 