I have the following function
$$f(\gamma,\kappa)=\gamma\kappa M^{2}(h_{v}+2(\kappa+\gamma M-\gamma)h_{\epsilon})(\kappa-\gamma)$$
What I want to know is how does $f(.)$ vary when $(\kappa-\gamma)$ changes. What I'm thinking of is to find the partial derivation with respect to $(\kappa-\gamma)$:
$$\frac{\partial f(\gamma,\kappa)}{\partial(\kappa-\gamma)}=?$$
I am quite sure, i cannot just do the following:
$$\frac{\partial f(\gamma,\kappa)}{\partial(\kappa-\gamma)}=\gamma\kappa M^{2}(h_{v}+2(\kappa+\gamma M-\gamma)h_{\epsilon})$$ How can I solve this Problem?
Assuming that the only variables ar $\gamma$ and $\kappa$: you must make a change of variables, for example: $$ \left\{ \begin{array}{c} x=\gamma+\kappa\\ y=\kappa-\gamma \end{array} \right. $$ Since $2\gamma=x-y$ and $2\kappa=x+y$, you can write $$ g(x,y)=\frac{1}{4}(x+y)(x-y) M^{2}(h_{v}+2(y+\frac{x-y}{2} M)h_{\epsilon})y $$ Now you can make the partial derivative of $g$ respect to $y$ and write again in the variables $\gamma$ and $\kappa$.