How to differentiate (ln n) ^(ln n) f(n) = (ln n)^(ln n) Can someone explain me how to differentiate the above function? I am trying the following solution f'(n) = (ln n((ln n) ^ (ln n) - 1))/((1/n ln 2))

Madison Costa

Madison Costa

Answered question

2022-11-11

How to differentiate ( ln n ) ln n ?
f ( n ) = ( ln n ) ln n
Can someone explain me how to differentiate the above function?
I am trying the following solution
f ( n ) = ln n ( ( ln n ) ln n 1 ) ( 1 n ln 2 )

Answer & Explanation

meexeniexia17h

meexeniexia17h

Beginner2022-11-12Added 18 answers

Let f ( x ) = ( log x ) log x ; then
log f ( x ) = log x log log x
and so, differentiating both sides,
f ( x ) f ( x ) = 1 x log log x + log x 1 log x 1 x = 1 + log log x x
If f ( x ) = ( log a x ) log a x , then we have again
log f ( x ) = log a x log log a x
(no subscript means natural logarithm). So
f ( x ) f ( x ) = 1 x log a log log a x + log a x 1 log a x 1 x log a = 1 + log log a x x log a
remembering that if g ( x ) = log a x, then g ( x ) = 1 / ( x log a ).
kunguwaat81

kunguwaat81

Beginner2022-11-13Added 2 answers

Let y = u v and u = v = ln ( x ). It then follows that
d y d x = y u d u d x + y v v u d x = v u v 1 x + ln ( u ) u v x = ln ( x ) ln ( x ) ( 1 + ln ( ln ( x ) ) ) x
Which follows from the multi-variable chain rule.

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