Product_quotient

# Product_quotient - PRODUCT AND QUOTIENT RULES Suppose that...

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PRODUCT AND QUOTIENT RULES Suppose that you make money by manufacturing chalks. To make more money, you can increase the speed or the amount of time of making chalks. If you do both, how does your income change? Let f be the amount of chalks you can make. If we let g be the speed of making chalks and h be the amount of time, we can see that f = gh . When we increase g , even if h remains the same, f also increases and the increment should be proportional to the increment of g . It suggests that f 0 contains a term g 0 h . The same argument holds for the increment of h , thus we can guess f 0 consists of g 0 h and gh 0 . We can prove this formula as follows. d dx [ f ( x ) g ( x )] = lim h 0 f ( x + h ) g ( x + h ) - f ( x ) g ( x ) h = lim h 0 ± f ( x + h ) g ( x + h ) - f ( x + h ) g ( x ) h + f ( x + h ) g ( x ) - f ( x ) g ( x ) h ² = lim h 0 f ( x + h ) g ( x + h ) - g ( x ) h + lim h 0 g ( x ) f ( x + h ) - f ( x ) h = f ( x ) g 0 ( x ) + f 0 ( x ) g ( x ) Example 1. Diﬀerentiate f ( x ) = x

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## This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.

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Product_quotient - PRODUCT AND QUOTIENT RULES Suppose that...

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