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# lec08 - 7 Dierentiation Rules P K Lamm Lecture Notes(17:33...

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7. Differentiation Rules P. K. Lamm 09/27/11 (17:33) p. 1 / 6 Lecture Notes: 7. Differentiation Rules These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1. Rules of Differentiation After some experience with computing derivatives using the definition of the derivative, the next theorems may come as some relief. Theorem: If f ( x ) = k , for any constant k and all x , then f 0 ( x ) = 0. Proof: Using the definition of the derivative, f 0 ( x ) = lim h 0 f ( x + h ) - f ( x ) h = lim h 0 k - k h = 0 . Theorem: If n > 0 is an integer and f ( x ) = x n , then f 0 ( x ) = nx n - 1 . Proof: The proof is not hard for cases n = 1 and n = 2, which we show first below: For n = 1, f ( x ) = x , and f 0 ( x ) = lim h 0 f ( x + h ) - f ( x ) h = lim h 0 ( x + h ) - x h = lim h 0 h h = 1 = 1 · x 0 . For n = 2, f ( x ) = x 2 , and f 0 ( x ) = lim h 0 ( x + h ) 2 - x 2 h = lim h 0 x 2 + 2 xh + h 2 - x 2 h = lim h 0 h (2 x + h ) h = lim h 0 2 x + h = 2 x = 2 x 1 . 1

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7. Differentiation Rules P. K. Lamm 09/27/11 (17:33) p. 2 / 6 For general n and f ( x ) = x n , we use the Binomial Formula to write ( x + h ) n = x n + nx n - 1 h + n ( n - 1) 2 x n - 2 h 2 + · · · + nxh n - 1 + h n . Then f 0 ( x ) = lim h 0 ( x + h ) n - x n h = lim h 0 x n + nx n - 1 h + n ( n - 1) 2 x n - 2 h 2 + · · · + nxh n - 1 + h n - x n h = lim h 0 h nx n - 1 + n ( n - 1) 2 x n - 2 h + · · · + nxh n - 2 + h n - 1 h = lim h 0 nx n - 1 + n ( n - 1) 2 x n - 2 h + · · · + nxh n - 2 + h n - 1 !
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lec08 - 7 Dierentiation Rules P K Lamm Lecture Notes(17:33...

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