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


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