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