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dokezwa17 2022-05-27 Answered
Derivative of ( ln x ) ln x
How can I differentiate the following function?
f ( x ) = ( ln x ) ln x .
Is it a composition of functions? And if so, which functions?
Thank you.
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Answers (2)

rideonthebussp
Answered 2022-05-28 Author has 10 answers
The function f can be expressed as
f ( x ) = ( ln x ) ln x = e ( ln x ) ln ( ln x ) .
Hence f ( x ) = exp ( g ( x ) ) , where g ( x ) = ( ln x ) ln ( ln x ), and thus
f ( x ) = ( e ( ln x ) ln ( ln x ) ) = e ( ln x ) ln ( ln x ) ( ( ln x ) ln ( ln x ) ) = ( ln x ) ln x ( 1 x ln ( ln x ) + ln x 1 x ln x ) .
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Anthony Kramer
Answered 2022-05-29 Author has 2 answers
Note that
a b = ( e ln a ) b = e b ln a
so
( ln x ) ln x = ( e ln ln x ) ln x = e ln x ln ln x = x ln ln x
either of which which you should be able to do with methods you already know.
We can apply this technique generally to calculate the derivative of f ( x ) g ( x ) :
f ( x ) g ( x ) = e g ln f
so
d d x f g = d d x e g ln f = ( d d x ( g ln f ) ) e g ln f (chain rule) = ( d d x ( g ln f ) ) f g = ( g ln f + g f f ) f g (product rule)
where f = d f d x and g = d g d x
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