# Thread: i need some help deriving these log functions

1. ## i need some help deriving these log functions

i looked at the steps, and the ones i found don't make sense to me. my teacher only allows us to use properties that we have learned previously for each problem
anyways, i cant figure out how to do these at all, so if you could point me in the right direction of how to solve these in the most basic way...

i need some help deriving these log functions:

f(x)= (log x)^x

f(x)=x^ log x

2. a) for $\displaystyle f(x)=x\cdot \ln x$ You have to apply the 'product rule' ...

$\displaystyle f(x)= u(x)\cdot v(x) \rightarrow f^{'}(x)= u(x)\cdot v^{'}(x) + v(x)\cdot u^{'} (x)$ (1)

b) for $\displaystyle f(x)= x^{\ln x}$ You have to apply the rule...

$\displaystyle f(x) = u\{v(x)\} \rightarrow f^{'} (x) = \frac{du}{dv} \cdot v^{'} (x)$ (2)

... remembering that...

$\displaystyle x^{\ln x} = e^{\ln^{2} x}$ (3)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. damn..im terrible sorry, but i meant to type (log x)^x for the first one
apparently, some keys of my keyboard arent what they should be

that one was and still is causing me heart ache

4. All right!... for $\displaystyle f(x)= (\ln x)^{x}$ You have to apply again the rule...

$\displaystyle f(x)= u \{v(x)\} \rightarrow f^{'}(x) = \frac{du}{dv}\cdot v^{'}(x)$

... remembering that...

$\displaystyle (\ln x)^{x} = e^{x\cdot \ln \ln x}$

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$