ep7 - 92.131 Calculus 1 Logarithmic Differentiation...

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92.131 Calculus 1 Logarithmic Differentiation Determine the derivatives of the following functions: 1) x x x f sin 2 ) 1 ( ) ( + = 2) x x x x x f sin 1 ) 1 ( ) ( 5 4 10 2 + + = 3) x x x f ln ) ( = 4) 1 2 ) (sec ) ( + = x x x f 5) 2 ) (sin ) ( x x x f = 6) x x x x f 2 4 2 sin ) 1 ( 1 ) ( + + = 7) x x x f sin ) (cos ) ( =
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92.131 Calculus 1 Logarithmic Differentiation Solutions 1 ) x x x f sin 2 ) 1 ( ) ( + = Using logarithmic differentiation, log both sides and simplify the RHS! ) 1 ln( sin ] ) 1 ln[( )] ( ln[ 2 sin 2 + = + = x x x x f x Now differentiate both sides wrt x , and we get 1 sin 2 ) 1 ln( cos ) ( ) ( 2 2 + + + = x x x x x x f x f , and hence + + + + = + + + = 1 sin 2 ) 1 ln( cos ) 1 ( 1 sin 2 ) 1 ln( cos ) ( ) ( 2 2 sin 2 2 2 x x x x x x x x x x x x f x f x 2) x x x x x f sin 1 ) 1 ( ) ( 5 4 10 2 + + = . ) ln(sin ln 5 1 ) 1 ln( 2 1 ) 1 ln( 10 sin 1 ) 1 ( ln )] ( ln[ 4 2 5 4 10 2 x x x x x x x x x f + + + = + + = , and so
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ep7 - 92.131 Calculus 1 Logarithmic Differentiation...

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