Answer
\[
\begin{array}{l}
6 \frac{d w}{d u}=\frac{\partial w}{\partial z}+\frac{\partial w}{\partial y}+\frac{\partial w}{\partial x}
\end{array}
\]
Work Step by Step
Let $f$ be a differential function of one variable
\[
x+2 y+3 z=u
\]
Let $f(u)=w$
\[
\begin{array}{l}
\frac{\partial w}{\partial x}=\frac{d w}{d u} \frac{\partial u}{\partial x} \\
\frac{\partial w}{\partial y}=\frac{d w}{d u} \frac{\partial u}{\partial y} \\
\frac{\partial w}{\partial z}=\frac{d w}{d u} \frac{\partial u}{\partial z}
\end{array}
\]
Let's consider $\frac{\partial u}{\partial z}=3, \quad \frac{\partial u}{\partial y}=2 \quad$ and $\quad \frac{\partial u}{\partial x}=1$
with $\quad \frac{\partial w}{\partial x}=1 f^{\prime}(u), \quad \frac{\partial w}{\partial y}=2 f^{\prime}(u) \quad$ and $\quad \frac{\partial w}{\partial z}=3 f^{\prime}(u)$
Thus, here we can write
\[
\begin{array}{l}
1 \int^{\prime}(u)+2 f^{\prime}(u)+3 f^{\prime}(u) =\frac{\partial w}{\partial x}+\frac{\partial w}{\partial y}+\frac{\partial w}{\partial z}\\
\frac{\partial w}{\partial x}+\frac{\partial w}{\partial y}+\frac{\partial w}{\partial z}=6 f^{\prime}(u) \\
6 \frac{d w}{d u}=\frac{\partial w}{\partial z}+\frac{\partial w}{\partial y}+\frac{\partial w}{\partial x}
\end{array}
\]
