Multi-variable chain rule with multi-variable functions as arguments What is the chain rule of a multi-variable function with arguments that are also multi-variable functions?
Suppose $x$, $y$, $z$ are independent variables. I mean changing $x$ won't change $y$ and $z$.
Is the general form of multi-variable chain rule similar to the following?
$$\frac{\partial w(u(x, y), v(y, z), t(y, z))}{\partial x} = {\frac{\partial w}{\partial u}}\cdot{\frac{\partial u}{\partial x}} + {\frac{\partial w}{\partial v}}\cdot{\frac{\partial v}{\partial x}} $$
Thanks.

\begin{align}
\frac{\partial w(u(x, y), v(y, z), t(y, z))}{\partial x} &= {\frac{\partial w}{\partial u}}\cdot{\frac{\partial u}{\partial x}} + {\frac{\partial w}{\partial v}}\cdot\overbrace{\frac{\partial v}{\partial x}}^0+ {\frac{\partial w}{\partial t}}\cdot\overbrace{\frac{\partial t}{\partial x}}^0 \\
&={\frac{\partial w}{\partial u}}\cdot{\frac{\partial u}{\partial x}},
\end{align}
or what I think you meant to write
\begin{align}
\frac{\partial w(u(x, y), v({\color{red}x}, z), t(y, z))}{\partial x} &= {\frac{\partial w}{\partial u}}\cdot{\frac{\partial u}{\partial x}} + {\frac{\partial w}{\partial v}}\cdot{\frac{\partial v}{\partial x}}+ {\frac{\partial w}{\partial t}}\cdot\overbrace{\frac{\partial t}{\partial x}}^0 \\
&={\frac{\partial w}{\partial u}}\cdot{\frac{\partial u}{\partial x}}+ {\frac{\partial w}{\partial v}}\cdot{\frac{\partial v}{\partial x}}.
\end{align}
