First, in order to turn this function into something more manageable with more familiar rules, we must switch it around into an equivalent function. So, we turn #(lnx)^(lnx)# into #e^((ln(lnx))(lnx))#. This works because, due to raising e to any power is an inverse function to taking the natural log of any number, #e^ln(x)=x# for any x. This may seem like a more complicated function we have made to take the derivative, but trust me, it makes thins simpler. So, since we know that the derivative of #e^u# is #u'*e^u# we'll take the derivative of #e^((ln(lnx))(lnx))# using this rule.
This means we must take the derivative of the exponent and multiply by the original function. This brings us a whole new, but much simpler problem. Now we must find the derivative of #ln(lnx)*lnx# using the product rule which means that the derivative of any two functions multiplied is the first function's derivative multiplied by the second function plus the first function multiplied by the second function's derivative.
Side note: since the derivative of #ln(u)# is #(u')/u# the derivative of #ln(lnx)# is #1/lnx# multiplied by the derivative of #lnx# which is #1/x#. This means the derivative of #ln(lnx)# is #1/(x*lnx)#.
This gives us the derivative of #ln(lnx)*lnx# which is #lnx/(x*lnx)+ln(lnx)/x#. If we do some cancellation we get: #1/x+ln(lnx)/x#, but since they both have denominators of x we can combine them to get #(ln(lnx)+1)/x#. THIS is the derivative of the original exponent which we will multiply with the original function.
This gives us: #((ln(lnx)+1)/x)e^(ln(lnx)*lnx# which can also be written as #((ln(lnx)+1)/x)(lnx)^(lnx)# and THIS is the final answer to the derivative of #(lnx)^lnx#.
