Statistical Thermodynamics and Rate Theories/Tips for deriving relations with partition functions

Use the rules of logarithms

You can use the rules of logarithms to isolate terms.

${\textrm {ln}}(ab)={\textrm {ln}}(a)+{\textrm {ln}}(b)$

${\textrm {ln}}\left({\frac {a}{b}}\right)={\textrm {ln}}(a)-{\textrm {ln}}(b)$

Derivatives of Constants are Equal to Zero

If you are taking a derivative of a function, you can use the rules of logarithms to isolate a term that depends on that variable. Now, you only need to take the derivative on this term. the derivatives of all other terms will be zero

${\frac {\partial {\textrm {ln}}(xy)}{\partial x}}={\frac {\partial {\textrm {ln}}(x)}{\partial x}}+{\frac {\partial {\textrm {ln}}(y)}{\partial x}}$

${\frac {\partial {\textrm {ln}}(xy)}{\partial x}}={\frac {\partial {\textrm {ln}}(x)}{\partial x}}+{\frac {\partial {\textrm {ln}}(y)}{\partial x}}$

${\frac {\partial {\textrm {ln}}(xy)}{\partial x}}={\frac {\partial {\textrm {ln}}(x)}{\partial x}}+0$

${\frac {\partial {\textrm {ln}}(xy)}{\partial x}}={\frac {\partial {\textrm {ln}}(x)}{\partial x}}$