Assume that you have functions/parameters
$\theta,\phi,\mu,p_1,\ldots,p_n$
We have functions
$\theta(\mu,p_1,\ldots,p_N)=f(\mu)$
$\phi(\mu,p_1,\ldots,p_N)$.
Notice that the first function only depend on $\mu$.
We also have the functions
$\mu(\theta,\phi)=g(\theta)$
$p_1(\theta,\phi),\ldots,p_N(\theta_,\phi)$.
Notice that the first function only depends on $\theta$.
We assume that all the partial derivatives exist.
I am wondering if it can be shown that $\phi$ does not depend on $\mu$, or said another way that $\frac{\partial \phi}{\partial \mu}=0$?