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2.3 Rules for Differentiation Colorado National Monument Vista High, AB Calculus. Book Larson, V9 2010Photo by Vickie Kelly, 2003
If the derivative of a function is its slope, then for a constant function, the derivative must be zero. example: The derivative of a constant is zero.
If we find derivatives with the difference quotient: (Pascal’s Triangle) We observe a pattern: …
examples: power rule We observe a pattern:…
examples: constant multiple rule: When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.
(Each term is treated separately) constant multiple rule: sum and difference rules:
Example: 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.)
First derivative (slope) is zero at:
product rule: Notice that this is not just the product of two derivatives. This is sometimes memorized as:
quotient rule: or
Higher Order Derivatives: 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. We will learn later what these higher order derivatives are used for.