Colorado National Monument Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 AP Calculus AB/BC 3.3 Differentiation Rules, p. 116

If the derivative of a function is its slope, then for a constant function, the derivative must be zero. Example 1: The derivative of a constant is zero. Constant Rule, p. 116

This is part of a pattern. Examples 2a & 2b: Power Rule, p. 116 The Power Rule works for both positive and negative powers

Example 3 Constant Multiple Rule, p. 117 where c is a constant and u is a differentiable function of x.

(Each term is treated separately) Sum and Difference Rules p. 117 where c is a constant and u is a differentiable function of x. where u and v are differentiable functions of x. Constant Multiple Rule, p. 117 Example 4aExample 4b

Example 5 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.)

Example 5 (cont.)

First derivative (slope) is zero at: Example 5 (cont.)

Product Rule, p. 119 Notice that this is not just the product of two derivatives. This is sometimes memorized as: Example 6:

Quotient Rule, p. 120 or Example 7: v u′u′ v′v′ u v2v2

Example 8 Find the tangent’s to Newton’s Serpentine at the origin and at the point (1, 2).

Example 8 (cont.) Next, find the slopes at the given points. For the point (0, 0), plug in 0 for x. y′ will be the slope. So, m = 4 and the equation for the tangent is y = 4x since the y-intercept is 0. For the point (1, 2), plug in 1 for x. So, m = 0 and the equation for the tangent is y = 2 since the m is 0.

Example 9: Find the first four derivatives of

Higher Order Derivatives, p.122 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.