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Math 131 Quiz 4 October 29, 2009

1) [5 points] Find the derivative and simplify.

-d- ( 2 ) dx log2 x - 1 .

Solution. By the chain rule,

-d- ( 2 ) dx log2 x - 1 1 1 ( )′ = -----2----- x2 - 1 ln 2x - 1 -1-----1--- = ln 2x2 - 1 2x = ------2x----- ln 2 ⋅ (x2 - 1)

2) [5 points] Use implicit differentiation to find y′ and evaluate y′ at the indicated point.

x2 - y = ln y; (1,1).

Solution.

d-(x2 - y) = -d-lny dx dx d d d --x2 - ---y = ---lny dx dx dx dy- 1-dy- 2x - dx = y dx 2x = dy-+ 1dy- dx ydx ( ) 2x = dy- 1 + 1- dx y dy 2x --- = ----1- dx 1 + y Therefore, | dy-|| --2-- dx | = 1 + 1 = 1 (1,1)