Find the derivative of the following function: y(x)=(ln(2x))^(5x),x>1/2

Pellagra3d 2022-10-24 Answered
Find the derivative of the following function: y ( x ) = ( ln ( 2 x ) ) 5 x , x > 1 2
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Answers (1)

cdtortosadn
Answered 2022-10-25 Author has 19 answers
Whenever you have a function of the form y = f ( x ) g ( x ) , it is usually a good idea to use logarithmic differentiation:
log y = g ( x ) log f ( x ) ,
hence
d d x [ log y ] = 1 y d y d x = d d x [ g ( x ) log f ( x ) ] ,
and
d y d x = f ( x ) g ( x ) ( g ( x ) f ( x ) d f d x + d g d x log f ( x ) ) .
So, in your case, f ( x ) = log 2 x and g ( x ) = 5 x, so we have
d f d x = 1 2 x 2 = 1 x ,
and
d g d x = 5 ,
therefore after minor simplifications,
d y d x = ( log 2 x ) 5 x ( 5 log 2 x + 5 log ( log 2 x ) ) .
The only additional step that was done was to put the fractions over a common denominator of log 2 x and factoring:
d y d x = 5 ( log 2 x ) 5 x 1 ( 1 + ( log 2 x ) log ( log 2 x ) ) .
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