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# Mock Gateway 2 Answers - Mock Gateway II(Derivatives 1 For...

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Mock Gateway II (Derivatives) 1. For f ( x ) = 1 /x , we have: lim h 0 f ( x + h ) - f ( x ) h = lim h 0 1 x + h - 1 x h = lim h 0 ( 1 x + h - 1 x ) x ( x + h ) hx ( x + h ) = lim h 0 x - ( x + h ) hx ( x + h ) = lim h 0 - h hx ( x + h ) = lim h 0 - 1 x ( x + h ) = - 1 x ( x + 0) = - 1 x 2 For f ( x ) = x 2 , we have: lim h 0 f ( x + h ) - f ( x ) h = lim h 0 ( x + h ) 2 - x 2 h = lim h 0 x 2 + 2 xh + h 2 - x 2 h = lim h 0 2 xh + h 2 h = lim h 0 (2 x + h ) = 2 x + 0 = 2 x For f ( x ) = x 3 , we have:

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lim h 0 f ( x + h ) - f ( x ) h = lim h 0 ( x + h ) 3 - x 3 h = lim h 0 x 3 + 3 x 2 h + 3 xh 2 + h 3 - x 3 h = lim h 0 3 x 2 h + 3 xh 2 + h 3 h = lim h 0 3 x 2 + 3 xh + h 2 = 3 x 2 + 3 x (0) + 0 2 = 3 x 2 For f ( x ) = x , we have: lim h 0 f ( x + h ) - f ( x ) h = lim h 0 x + h - x h = lim h 0 ( x + h - x )( x + h + x ) h ( x + h + x ) = lim h 0 ( x + h ) - x h ( x + h + x ) = lim h 0 h h ( x + h + x ) = lim h 0 1
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Mock Gateway 2 Answers - Mock Gateway II(Derivatives 1 For...

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