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.

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

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

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

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

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

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

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