Consider the following non-linear differential equation $$\label{star} \dot{x}(t) = f(x(t))+g(x(t),u(t)), \quad t\ge 0, \ x(0)=x_0\in\mathbb{R},\tag{$\star$} $$ where $f(\cdot)$ and $g(\cdot)$ are smooth functions of their arguments and $u(t)$, $t\ge 0$ is an input function. Suppose that the input dynamics satisfy another differential equation $$\label{u} \dot{u}(t) = h(u(t)) + z(x(t)), \quad t\ge 0, \ u(0)=u_0\in\mathbb{R},\tag{$\#$} $$ where $h(\cdot)$ and $z(\cdot)$ are smooth functions.
Moreover suppose that $(x^*(t),u^*(t))=(0,\bar{u}(t))$, where $\dot{\bar{u}}(t) = h(\bar{u}(t))$, $\bar{u}(0)=u_0$, is an equilibrium for the system in \eqref{star}. The task is to linearize \eqref{star} around $(x^*(t),u^*(t))$.
My (perhaps silly) question. Is the following one the correct linearized dynamics? $$ \dot{\delta_x}(t) = \frac{\mathrm{d} f(x)}{\mathrm{d} x}\bigg|_{x=0}\delta_x(t)+\frac{\partial g(x,u)}{\partial x}\bigg|_{x=0,u=\bar{u}}\delta_x(t)+\frac{\partial g(x,u)}{\partial u}\bigg|_{x=0,u=\bar{u}}\delta_u(t). $$ If so, since the dynamics of $u(t)$ depends on $x(t)$ from \eqref{u}, when I explicitly compute the derivatives $\frac{\partial g(x,u)}{\partial x}$ and $\frac{\partial g(x,u)}{\partial u}$ do I also need to evaluate the derivatives $\frac{\mathrm{d} u}{\mathrm{d} x}$ and $\frac{\mathrm{d} x}{\mathrm{d} u}$?