find the derivative
f(x)=x^(lnx)

Question

find the derivative
f(x)=x^(lnx)

hartnn · Answer

did you try taking natural log (ln) on both sides ??? as a first step ?

anonymous · Answer

how would it look like??

hartnn · Answer

$\Large \ln f(x) = \ln [x^{\ln x}]$

hartnn · Answer

now you can simplify right side using the property of log that says
$\ln a^b = b\ln a$

anonymous · Answer

okay how would i know en to put ln on both sides???

hartnn · Answer

when you have a function of x in the exponent! 
then its always preferable to take ln on both side first

here you had ln x in the exponent

anonymous · Answer

okay so it is ln^x lnx

hartnn · Answer

its 
$\Large \ln f(x) = \ln [x^{\ln x}]$

try simplifying right side

anonymous · Answer

$$
x^{\ln x} = (e^{\ln x})^{\ln x} = e^{(\ln x)^2}
$$

anonymous · Answer

Another way to approach this problem.

anonymous · Answer

ln f(x)=lnx(lnx) ??

anonymous · Answer

is it supposed to be squared?? for short??

hartnn · Answer

yes!,
that would be 
$\ln f(x) = (\ln x)^2$

now differentiate both sides, with respect to x 
use chain rule

anonymous · Answer

2x/x2 on the right side??

anonymous · Answer

2x/x^2

anonymous · Answer

what will the left be??

hartnn · Answer

$[(\ln x)^2]' = 2 \ln x (\ln x)'$
isn't it ??

anonymous · Answer

so its not the exponent that gets distributed??

hartnn · Answer

that was just a simple implementation of chain rule
d/dx (x^2) = 2x
so d/dx (ln x)^2 = 2 ln x d/dx(ln x)
can you simplify that ?

anonymous · Answer

oh i see waht you did

anonymous · Answer

i had thought you put (lnx)^2 as the left side of the equation

hartnn · Answer

so, you know d/dx (ln x) ?

anonymous · Answer

1/x ???

hartnn · Answer

yes,
so your right side will look like this
$\dfrac{2\ln x}{x}$

hartnn · Answer

the left side would be [ln f(x)]' 
apply chain rule here too ?
what would u get ?

anonymous · Answer

f'(x)/f(x)

hartnn · Answer

correct!

so, finally your f'(x) will be 

$f(x)\dfrac{2\ln x}{x} $ 
right?! 
which is same as 
$\Large f'(x) = x^{\ln x} \times \dfrac{2\ln x}{x}$
thats it!

hartnn · Answer

ask if any doubts :)

anonymous · Answer

why did you put x^lnx

anonymous · Answer

on the f'(x)=

hartnn · Answer

no... i put x^ln x for f(x)
that f'(x) is still on the left

anonymous · Answer

oh okay i get it thank you

hartnn · Answer

welcome ^_^

anonymous · Answer

could you help me with a different problem please

hartnn · Answer

can you please post it as a new question ?
so that if i am unavailable, there are many others to help you out :)

anonymous · Answer

okay thank you