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solutions_10_09 - Math 1a Product/quotient rules and...

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Math 1a: Product/quotient rules and applications October 9, 2009 1. Determine the following derivatives. 1. d dt ( te t ) = e t + te t . 2. d ds ( s 3 2 s ) = 3 s 2 2 s + s 3 2 s log(2). 3. d dp p 2 p 2 + 1 = 2 p - p 2 ( p 2 + 1) 2 . 2. Differentiate the following functions of x in two different ways. Check that your answers agree. Which way was easier? 1. a x b x . By the product rule, d dx ( a x b x ) = a x b x log( b ) + a x b x log( a ) = a x b x (log( a ) + log( b ) = a x b x log( ab ) . Also, we have a x b x = ( ab ) x . So, d dx ( ab ) x = ( ab ) x log( ab ) . 2. ( x 3 + 2)(4 x 2 - 1) One can either use the product rule, or multiply first and then differentiate the resulting polynomial. 3. 4 x + 3 x 2 x One can use the quotient rule, or simplify as 4 x + 3 x 2 x = 2 x + (3 / 2) 3 , and then use the sum rule. 3. On what interval is the function f ( x ) = x 3 e x increasing? Solution. We know that f ( x ) is increasing on the intervals where f 0 ( x ) is positive. By the product rule, we have f 0 ( x ) = x 3 e x + 3 x 2 e x = e x x 2 ( x + 3). Note that this is positive for - 3 < x < 0 and 0 < x .
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