Write out the chain rule for $\frac{\partial h}{\partial x}$, where $h(u,v) = g(u(x,y,z), v(x,y,z))$.
I think it's $\frac{\partial g}{\partial u} \cdot \frac{\partial u}{\partial x} + \frac{\partial g}{\partial v} \cdot \frac{\partial v}{\partial x}$.
Use this to calculate the partial derivative with respect to $x$ of $\textbf{D} (g \circ f)$ at $(0, 1, 0)$ where $g(u,v) = (e^u, u + \sin v)$ and $f(x,y,z) = (xy, yz)$.
So first I calculate the partial derivative of $g$ w.r.t $u$ according to my previous answer (the first step is to obtain $\frac{\partial g}{\partial u} \cdot \frac{\partial u}{\partial x}$), so:
$$\left( \begin{array}{cc} e^u & 0 \\ 1 & \cos v \\\end{array} \right)$$
However, this is the entire derivative. How do I write out the partial derivative of $g$ with respect to $u$ only without a matrix?