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Unformatted text preview: Chapter 3. Di ff erentiation 3.3 Di ff erentiation Rules Derivative of a Constant Function. If f has the constant value f ( x ) = c , then df dx = d dx [ c ] = 0 . Proof. From the definition: f ( x ) = lim h f ( x + h )- f ( x ) h = lim h o c- c h = lim h 0 = 0 . QED Power Rule for Positive Integers If n is a positive integer, then d dx [ x n ] = nx n- 1 . Note. Before we present the proof of the Power Rule, we introduce the Binomial Theorem. Theorem. Binomial Theorem Let a and b be real numbers and let n be a positive integer. Then ( a + b ) n = a n + na n- 1 b + n ( n- 1) 2 a n- 2 b 2 + . . . + nab n- 1 + b n = n i =0 n i a n- i b i where n i = n ! ( n- i )! i ! and i ! = ( i )( i- 1)( i- 2) (3)(2)(1) . Note. We can prove the Binomial Theorem using Mathematical Induc- tion ....
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