# Partial derivation with respect to a difference

I have the following function

f(γ,κ)=γκM2(hv+2(κ+γM−γ)hϵ)(κ−γ)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(.)f(.) vary when (κ−γ)(\kappa-\gamma) changes.
What I’m thinking of is to find the partial derivation with respect to (κ−γ)(\kappa-\gamma):

∂f(γ,κ)∂(κ−γ)=?\frac{\partial f(\gamma,\kappa)}{\partial(\kappa-\gamma)}=?

I am quite sure, i cannot just do the following:

∂f(γ,κ)∂(κ−γ)=γκM2(hv+2(κ+γM−γ)hϵ)\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?

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So: MM, hvh_v, and hϵh_{\epsilon} are constants? And the domain of ff is all pairs of real numbers?
– 6005
2 days ago

Yes to both questions
– Soloto
2 days ago

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Assuming that the only variables ar γ\gamma and κ\kappa:
you must make a change of variables, for example:
{x=γ+κy=κ−γ
\left\{
\begin{array}{c}
x=\gamma+\kappa\\
y=\kappa-\gamma
\end{array}
\right.

Since 2γ=x−y2\gamma=x-y and 2κ=x+y2\kappa=x+y, you can write
g(x,y)=14(x+y)(x−y)M2(hv+2(y+x−y2M)hϵ)y
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 gg respect to yy and write again in the variables γ\gamma and κ\kappa.