How to find $\bigg(\frac{\partial P}{\partial T}\bigg)_q$, given $\bigg(\frac{\partial \big(\ln (P) \big)}{\partial T}\bigg)_q$

Question

How can I find, partial derivative of $P$ with respect to $T$ at a given $q$ $$\bigg(\frac{\partial P}{\partial T}\bigg)_q$$ given that I know, partial derivative of $\ln P$ with respect to $T$ at a given $q$ $$\bigg(\frac{\partial \big(\ln (P) \big)}{\partial T}\bigg)_q$$

Thanks so much.

Answer 1

For a smooth function $f$ you have: $$ \frac{d\log{f(x)}}{dx}=\frac{f'(x)}{f(x)} \Rightarrow f'(x)=f(x) \frac{d\log{f(x)}}{dx} $$ With you notation this gives: $$ \left(\frac{\partial P}{\partial T}\right)_q=P(T,q)\left(\frac{\partial \log{P}}{\partial T}\right)_q $$

Answer 2

Notice that:

$$\frac{\partial \big(\ln (P) \big)}{\partial T} = \frac{1}{P}\frac{\partial P }{\partial T},$$

or equivalently: $$\frac{\partial P }{\partial T} = P \frac{\partial \big(\ln (P) \big)}{\partial T}.$$

Moreover, notice that you don't need to care about $q$ since it is assumed constant in both the expressions.