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Unit 2 – Differentiation. Section 2.3 Product & Quotient Rules and Higher-Order Derivatives. Objectives: Find the derivative of a function using the Product Rule. Find the derivative of a function using the Quotient Rule. Find the derivative of a trigonometric function. - PowerPoint PPT Presentation

Section 2.3 Product & Quotient Rules and Higher-Order Derivatives

Section 2.3Product & Quotient Rules and Higher-Order DerivativesUnit 2 DifferentiationObjectives:

Find the derivative of a function using the Product Rule.Find the derivative of a function using the Quotient Rule.Find the derivative of a trigonometric function.Find a higher-order derivative of a function.Theorem 2.7The Product RuleExample 1Example 1 continuedSince the two function pieces were polynomials, we can also find the derivative without using the product rule.Example 2Example 3Here it is necessary to use the several rules together.Product RuleConstant Multiple RuleDifference RuleThis answer can be represented multiple ways, & when compared to an answer (like from the back of the book), it could be given differently.Example 3 continuedAlternative way to write the answer:Theorem 2.8The Quotient RuleExample 4Example 5Example 5 continuedExample 5 continuedExample 6First with the Quotient Rule.Example 6 continuedNow with the Constant Multiple Rule.Theorem 2.9Derivatives of Trigonometric FunctionsTo aid in memorizing these, remember the derivatives of the cofunctions (cosine, cotangent, and cosecant) all require negatives in front.Example 7It will always be to your advantage to memorize these rules for speed and ease in working problems. However, it is important to remember that even though you may forget a rule, they can easily be derived using the Quotient Rule & a few trigonometric identities.Example 7 continuedThe proofs of the other two derivatives are similar.Example 8Example 9Example 9 continuedNow to differentiate it in its other form:Higher-Order DerivativesPosition functionVelocity functionAcceleration functionHigher-Order DerivativesThe acceleration function is called the second derivative of the position function and is an exampled of a higher-order derivative. Below are how higher-order derivatives are denoted:Example 10Moon:Earth:Interpreting a Derivative from a GraphOnce again, one of the ways the AP exam will test the concept of a derivative rather than just your memorization of rules will be in how it asks you to interpret derivatives from a graph. The following example will illustrate one way on how this is achieved.Example 11

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Example 12AHere is an example of a particular multiple choice AP test question. This question would be on the non-calculator portion of the test. What it really wants us to find is an approximation using the average rate of change (the slope of the secant line).Example 12BAgain we want to find just the average rate of change, but we will find it between the x-values of 1.4 and 1.6.