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## AP Calculus AB/BC

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**Photo by Vickie Kelly, 2003**Greg Kelly, Hanford High School, Richland, Washington AP Calculus AB/BC 3.3 Differentiation Rules, p. 116 Colorado National Monument**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. Constant Rule, p. 116 Example 1:**Power Rule, p. 116**This is part of a pattern. Examples 2a & 2b: The Power Rule works for both positive and negative powers**Constant Multiple Rule, p. 117**Example 3 where c is a constant and u is a differentiable function of x.**Constant Multiple Rule, p. 117**where c is a constant and u is a differentiable function of x. Sum and Difference Rules p. 117 where u and v are differentiable functions of x. Example 4a Example 4b (Each term is treated separately)**Horizontal tangents occur when slope = zero.**Example 5 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.)**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: u′ u v v′ v2**Example 8**Find the tangent’s to Newton’s Serpentine at the origin and at the point (1, 2).**Example 8 (cont.)**For the point (1, 2), plug in 1 for x. 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 = 0 and the equation for the tangent is y = 2 since the m is 0. So, m = 4 and the equation for the tangent is y = 4x since the y-intercept is 0.**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, p.122 (y double prime) We will learn later what these higher order derivatives are used for. p