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Differentiation Formulas: Products and Quotients

# Differentiation Formulas: Products and Quotients - 18.01...

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18.01 Calculus Jason Starr Fall 2005 Lecture 3. September 13, 2005 Homework. Problem Set 1 Part I: (i) and (j). Practice Problems. Course Reader: 1E-1, 1E-3, 1E-5. 1. Another derivative. Use the 3-step method to compute the derivative of f ( x ) = 1 / 3 x + 1 is, f ( x x 3 / 2 / 2 . ) = 3(3 + 1) Upshot: Computing derivatives by the deFnition is too much work to be practical. We need general methods to simplify computations.

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± ± ± ± ± ± ± ± ² ³ ² ³ ³ ² ³ ² ³ ² ³ ² ³ ² ³ 18.01 Calculus Jason Starr Fall 2005 2. The binomial theorem. For a positive integer n , the factorial , n ! = n × ( n 1) × ( n 2) × ··· × 3 × 2 × 1 , is the number of ways of arranging n distinct objects in a line. For two positive integers n and k , the binomial coeﬃcient , n n ! n ( n 1) ··· ( n k + 2)( n k + 1) , = = k k !( n k )! 1 3 · 2 · k ( k 1) ··· is the number of ways to choose a subset of k elements from a collection of n elements. A funda- mental fact about binomial coeﬃcients is the following, n n n + 1 + = . k k k 1 This is known as Pascal’s formula . This link is to a webpage produced by MathWorld , part of Wolfram Research. The Binomial Theorem says that for every positive integer n and every pair of numbers a and b , ( a + b ) n equals, n n n a + na n 1 b + ··· + k a n k b k + ··· + nab n 1 + b .
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Differentiation Formulas: Products and Quotients - 18.01...

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