My problem comes from thermodynamics. I am trying to solve the Peng-Robinson EOS equation to get the residual enthalpy. Then I have an function $a(\operatorname{Tr})$ that is $$ a = \Bigl(1 + k \, \bigl( 1 - \sqrt{\operatorname{Tr}} \bigr) \Bigr)^2. $$ I need to solve $$ \frac{d\ln(a)}{d\ln(\operatorname{Tr})}. $$ Any idea on how to proced?
Once I know how to manage the logarithm part of the diferential I can solve using numeric methods.
let : $$y = \ln(Tr) \implies Tr = e^y $$ and
$$ ln(a)=ln\Bigl(1 + k \, \bigl( 1 - \sqrt{\operatorname{Tr}} \bigr) \Bigr)^2=2ln\Bigl(1 + k \, \bigl( 1 - \sqrt{\operatorname{e^y}} \bigr) \Bigr)$$
And therfore : $$\frac{d\ln(a)}{d\ln(\operatorname{Tr})}=\frac{d\ln(a)}{dy}=\frac{2d\ln\Bigl(1 + k \, \bigl( 1 - \sqrt{\operatorname{e^y}} \bigr) \Bigr)}{dy}=\frac{-k\sqrt{e^y}}{\Bigl(1 + k \, \bigl( 1 - \sqrt{\operatorname{e^y}} \bigr) \Bigr)}$$
Finally : $$\frac{d\ln(a)}{d\ln(\operatorname{Tr})}=\frac{-k\sqrt{Tr}}{\Bigl(1 + k \, \bigl( 1 - \sqrt{\operatorname{Tr}} \bigr) \Bigr)}$$