partial derivative with 2 variables

Question

I have a function f(x,y) with 2 variables x and y. I calculate the partial derivative and get the following equation $$\frac{\partial f (x,y)}{\partial x}=g(y)=0$$ To solve the problem $$g(y)=a+\frac{b}{log(1+y)}+\frac{c~y}{log(1+y)}=0$$ can I use the partial derivative ?$$\frac{\partial g (y)}{\partial y}=0$$