Silly question of vector field

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Updated: For $\frac{\partial U}{\partial y}line, $ why is $\frac{\partial g(y,z)}{\partial y} = y + h(z)$ but not zero + h(z)? (if it is$\frac{\partial }{\partial y} 1$...

Up-Updated: I get it now, I just have some confusion of diffentiation concept...(lol)

$\frac{\partial g(y,z)}{\partial y} = 1$ but as i am finding g(y,z), i shall $\int \:1\:dy$ to find the y (g(y)part) then I just need to find the amount of h(z).