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section37_material - Section 3.7 Derivatives of Powers Sums...

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MATH 110 BUSINESS CALCULUS CHAPTER 3 INTRODUCTION TO THE DERIVATIVE Section 3.7 Derivatives of Powers, Sums, and Constant Multiples Suppose we were to find a derivative of each of the following functions using the limit: 0 2 0 3 2 0 4 3 0 ( ) ( ) ( ) , ( ) lim 1 ( ) ( ) ( ) , ( ) lim 2 ( ) ( ) ( ) , ( ) lim 3 ( ) ( ) ( ) , ( ) lim 4 ... h h h h f x h f x f x x f x h f x h f x f x x f x x h f x h f x f x x f x x h f x h f x f x x f x x h + = = = + = = = + = = = + = = = See something interesting? A pattern? Once a pattern is recognized and established (proved), we can rely on it to find the derivatives of other functions. Thus, we develop a technique!
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Theorem: The Power Rule If n is any constant and ( ) n f x x = , then 1 ( ) n f x nx = Examples: 7 6 If ( ) , by the Power Rule we have ( ) 7 f x x f x x = = More Exercises: Find the derivative of each of the following function: 1. 10 ( ) f x x = 2. 2 1 ( ) f x x = 3. ( ) f x x = 4. 1 ( ) f x x = 5. ( ) 1 f x = 6. 4.8 ( ) f x x =
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Theorem: Derivatives of Sums, Differences, and Constant Multiples If ( ) f x and ( ) g x are any two differentiable functions, and if
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