Suppose I have a function $f(r, \theta, p_{\mu})$ in the phase space $(q^{\mu}, p_{\mu})$. I want to determine an expression for the partial derivative w.r.t $r$, i.e. $\frac{\partial f}{\partial r}$.
Is it true to say that,
$$ \frac{\partial f}{\partial r} =\frac{\partial f}{\partial \theta} \frac{\partial \theta}{\partial r} $$
?
My confusion comes from the $p_{\mu}$ dependence. Is the answer instead,
$$ \frac{\partial f}{\partial r} =\frac{\partial f}{\partial \theta} \frac{\partial \theta}{\partial r} +\frac{\partial f}{\partial p_{\mu}}\frac{\partial p_{\mu}}{\partial r} $$
? But then, since $\frac{\partial p_{\mu}}{\partial r} =0$, does this reduce to the first answer?