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Product & Quotient Rules Higher Order Derivatives Lesson 2.3 Slide 2 After this lesson, you should be able to: 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 trig function Find a higher-order derivative of a function Slide 3 Basic Rules Product Rule How would you put this rule into words? Slide 4 The derivative of a product of 2 functions is equal to the first times the derivative of the second plus the second function times the derivative of the first. Slide 5 Theorem 2.7 The Product Rule Slide 6 Try Some Use additional rules to determine the derivatives of the following function Slide 7 Try Some More Use additional rules to determine the derivatives of the following function a) Slide 8 Try Some More Use additional rules to determine the derivatives of the following function b) Slide 9 Try Some More Use additional rules to determine the derivatives of the following function b) Slide 10 Try Some More Use additional rules to determine the derivatives of the following function b) NOTE!!! YOU MUST FACTOR COMPLETELY!!!!!!!!!! GET IT DOWN TO THE POINT WHERE YOU COULD SOLVE FOR THE ZEROS OF THE EQUATION!!!!! Slide 11 Basic Rule Quotient Rule How would you put this rule into words? Slide 12 The derivative of a quotient of 2 functions is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all over the square of the denominator. Slide 13 Theorem 2.8 The Quotient Rule Slide 14 Just Checking... Find the derivatives of the given functions Slide 15 Just Checking... Find the derivatives of the given functions Slide 16 Just Checking... Find the derivatives of the given functions Slide 17 Just Checking... Find the derivatives of the given functions Slide 18 Slide 19 Slide 20 Slide 21 Slide 22 Slide 23 Slide 24 Slide 25 Slide 26 Slide 27 Other Trig Derivatives Now try it out Slide 28 Theorem 2.9 Derivatives of Trigonometric Function Slide 29 Other Trig Derivatives Slide 30 Slide 31 Slide 32 Slide 33 JUST AN EXAMPLE OF HOW COMPLICATED THEY CAN GET!!!! CAN WE SIMPLIFY FURTHER??? Slide 34 Derivatives of the remaining trig functions can be determined the same way. Slide 35 Find Those High Orders Find the requested derivatives Slide 36 Find each derivative. Slide 37 Find the derivative. Slide 38 Slide 39 Slide 40 Equation of a Horizontal Tangent Line Find the equations of the horizontal tangent lines for Slide 41 Equation of a Horizontal Tangent Line Find the equations of the horizontal tangent lines for Slide 42 Example: Write the equation of the tangent line at Equation of the Tangent Line Slide 43 Example: Write the equation of the tangent line at Equation of the Tangent Line