Suppose I have the function $$F(x,y) = x^a y^{1-a} = 1$$ where $a \in [0,1]$.
$$g(x,y) = - \frac{\partial F /\partial x}{\partial F / \partial y} = \frac{a}{1-a} \frac{y}{x} $$ $$\frac{\partial g}{\partial a} = -\frac{1}{(1-a)^2}\frac{y}{x}$$
So suppose I have the following set of graphs:
Here I just picked some values. Red is $a = \frac{1}{4}$, blue is $a = \frac{1}{2}$, black is $a = \frac{3}{4}$
I understand how the steepness and flatness of the different curves change as I vary $a$. But I am confused as to what $\frac{\partial g}{\partial a} $ tells me about the graph. In particular, $\frac{\partial g}{\partial a} $ is dependent on $a$. So even though I know $\frac{\partial g}{\partial a}$ tells me how much $g$ changes as I change $a$....changing $a$ changes $\frac{\partial g}{\partial a}$...so I am very confused!
My Question
What is $\frac{\partial g}{\partial a}$ telling me about the graph? Since $\frac{\partial g}{\partial a}$ is dependent on $a$, how does this affect things?