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

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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.

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

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examples: constant multiple rule:

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

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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) (So we expect to see two horizontal tangents, intersecting the curve at three points.) … and write tangent lines:

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

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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(uv) = v du + u dv = (2x 3 + 5x)(2x)+ (x 2 + 3)(6x 2 + 5) = (4x 4 + 10x 2 ) + (6x 4 + 23x 2 + 15) = 10x 4 + 33x 2 + 15 u = x 2 + 3 v = 2x 3 + 5x du/dx = dv/dx = 2x + 0 23x 2 + 5

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

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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 We will see later what these higher-order derivatives might mean in the real world! (y triple prime), the derivative of the second derivative.

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