Suppose I have a function $\rho(x(X,t),t)$, and I perform the partial derivative based on chain rule, what I have is $\frac{\partial \rho}{\partial t} = \frac{\partial\rho}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial \rho}{\partial t}$.
I am confused, both side has$\frac{\partial \rho}{\partial t}$.
On
The letter $\rho$ is being used to denote two different functions. Suppose that $R(u,t) = \rho(x(u,t),t)$. By the multivariable chain rule, $\frac{\partial R}{\partial t} = \frac{\partial \rho}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial \rho }{\partial t}$. For some reason the function $R$ is often called $\rho$, even though the name $\rho$ has already been used for a different function. This is a common abuse of notation and it frequently causes confusion.
The derivative on the left side shouldn’t be written using partial derivatives; rather, it is the total derivative. The term on the right side is the partial derivative in $t$ viewing $x$ as fixed.