derivative $f(z,y) = g(x+y)h(y)$ with respect to $y$


Suppose I have differentiable function $g(x+y)$ and $h(y)$ and define $z = x+y$. Also, I assume that for any $x_1,y_1$ there exists $x_2,y_2$ such that $g(x_1+y_1) = g(x_2+y_2)$. Then, I want to take derivate of $$f(z,y) = g(x+y)h(y)$$ with respect to $y$ and obtain the result like $$g(x+y)h'(y).$$ I want to know if it is possible. My argument is that whenever I change $y$, I can always find $x’$ to offset the effect of the change in $y$ on function $g$.