I'll use subscript notation for brevity, i.e. [tex]\frac{\partial f}{\partial x}=f_x[/tex].
By the chain rule,
[tex]z_u=z_xx_u+z_yy_u[/tex]
[tex]z_v=z_xx_v+z_yy_v[/tex]
[tex]z_w=z_xx_w+z_yy_w[/tex]
We have
[tex]z=x^4+xy^3\implies\begin{cases}z_x=4x^3+y^3\\z_y=3xy^2\end{cases}[/tex]
and
[tex]\begin{cases}x=uv^4+w^3\\y=u+ve^w\end{cases}\implies\begin{cases}x_u=v^4\\x_v=4uv^3\\x_w=3w^2\\y_u=1\\y_v=e^w\\y_w=ve^w\end{cases}[/tex]
When [tex]u=1,v=1,w=0[/tex], we have
[tex]\begin{cases}x(1,1,0)=1\\y(1,1,0)=2\end{cases}\implies\begin{cases}z_x(1,2)=12\\z_y(1,2)=12\end{cases}[/tex]
and the partial derivatives take on values of
[tex]\begin{cases}x_u(1,1,0)=1\\x_v(1,1,0)=4\\x_w(1,1,0)=0\\y_u(1,1,0)=1\\y_v(1,1,0)=1\\y_w(1,1,0)=1\end{cases}[/tex]
So we end up with
[tex]\boxed{\begin{cases}z_u(1,1,0)=24\\z_v(1,1,0)=60\\z_w(1,1,0)=12\end{cases}}[/tex]
