3.3 Differentiation Rules Colorado National Monument Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts Photo by Vickie Kelly, 2003

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

We saw that if , . This is part of a larger pattern! examples: “power rule”

constant multiple rule: examples:

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

Example: Find the horizontal tangents of: Horizontal tangents: (slope of y) = (derivative of y) = zero. Substitute these x values into the original equation to generate coordinate pairs: (0, 2), (-1, 1) and (1,1) … and write tangent lines: (So we expect to see two horizontal tangents, intersecting the curve at three points.)

First derivative y’ (slope) is zero at: …where y has slopes of zero The derivative of y is zero at x= -1, 0, 1…

product rule: Notice: the derivative of a product is not just the product of the two derivatives. (Our authors write this rule in the other order!) This rule can be “spoken” as: d(u∙v) = v ∙ du + u ∙ dv = (2x3 + 5x) (2x) + (x2 + 3) (6x2 + 5) = (4x4 + 10x2) + (6x4 + 23x2 + 15) = 10x4 + 33x2 + 15 u = x2 + 3 v = 2x3 + 5x du/dx = dv/dx = 2x + 0 2∙3x2 + 5

quotient rule: or u = 2x3 + 5x v = x2 + 3 du = dv =

Higher Order Derivatives: is the first derivative of y with respect to x. is the second derivative (“y double prime”), the derivative of the first derivative. is the third derivative (“y triple prime”), the derivative of the second derivative. We will see later what these higher-order derivatives might mean in “the real world!” p