Does the chain rule tell us anything about $\frac{d}{dx}h(z,y)$? Does the chain rule tell us anything about
$$\frac{d}{dx}h(z,y)?$$
Notation: the (real) variables $z$ and $y$ are understood to depend on $x,$ and $h$ is a differentiable function $\mathbb{R}^2 \rightarrow \mathbb{R}.$

If I understood it right you have the composition of a vectorial function $(z,y):\mathbb{R}\rightarrow\mathbb{R}^2$ and $h:\mathbb{R}^2\rightarrow\mathbb{R}$. If everything is sufficiently regular you have that 
$$
\frac{d}{dx}h(z,y)=\nabla h(z,y)\times\frac{d}{dx}(z,y)=\frac{\partial h}{\partial x_1}(z,y)z'+\frac{\partial h}{\partial x_2}(z,y)y',
$$
where $x_1$ and $x_2$ are the variables of $h$ as a function from $\mathbb{R}^2$ and $\times$ is a product of matrices (the grandient of $h$ is the 1x2 matrix of functions  $\nabla h=\left[ \frac{\partial h}{\partial x_1},\frac{\partial h}{\partial x_2}\right]$, and the differential of a vectorial function is defined component-wise).
This is a particular case of a general rule: given a couple of differentiable functions $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ and $g:\mathbb{R}^m\rightarrow\mathbb{R}^l$ then
$$
J(g\circ f)=Jg\circ f\times Jf
$$
where $Jf$ is the Jacobian matrix of $f$, defined by $(Jf)_{ij}:=\frac{\partial f_i}{\partial x_j}$ for $i=1,...,m$ and $j=1,...,n$. In particular if $n=m=l=1$ you have the usual chain rule for functions in one variable.
