Let's imagine, that we have a function $f$, that depends on $x,y$, it could be some nonlinear function, like $f(x,y,z..) = x^3 + x + y + y^2+ z^2 + z $ .. or $f(x,y,z) = x + z$ ... etc . I want to take the derivative of this function $f(x,y,z)$ with respect to (x+y), so $ \frac{\partial f(x,y,z)}{\partial(x+y)}$.
I cannot find the definition/transformation rule for this, but I am maybe just overlooking something.
EDIT:
The problem itself originates from a discreet quantum gravity approach (Causal Dynamical Triangulations). There is 4-dimensional spacetime approximated by a globally hyperbolic 3+1 dimensional foliated triangulation. As such, the triangulation is constructed with the help of 4 types of building blocks, s41, s32, s23, s14, where s is a "simplex" and ij shows the number of vertices at the one and the other slice of the foliation. The vertices of the triangulation lies on "t" integer slices. Let's say we have T number of slices, then a configuration is a foliated triangulation from $t = 0$ to $t = T$, with 4xT "fractional" slices ($\tau$), which will specify the location of each different simplex. E.G. s41(t = 0) is a simplex that has 4 vertices at slice 0 and 1 vertex at slice 1. s23(t+2) is a simplex that has 2 vertices at slice t+2 and 3 at slice t+3. $\tau = 4\cdot t + i$, $i = 0$ for s41, $i = 1 $ for s32 and $i = 2$ for s23 and $i = 3$ for s14.
Let's assume that we are in Euclidean space and all triangles are equilateral. The model is quite simple but it leads to a rich phase diagram. Without going into many details, there is a phase, where the shape (number of simplices in the function of $t$ or $\tau)$ of the triangulation can be described by a minisuperspace model, where there is only one parameter: the volume. The volume $n$is classically given by the number of spatial tetrahedra, thus $n = \sum_t n_t$, where $n_t$ signs all the s41 simplices at fixed integer slice of the foliation. Using this as the definition of the volume one can write the Effective action of the model, which is given by one variable: volume (it's like the scale factor in cosmology).
$$S_{eff}[n_t] = \frac{1}{\Gamma}\sum_t \frac{(n_t - n_{t+1})^2}{n_t + n_{t+1}} + \mu n_t^\gamma + \lambda n_t$$.
The quadratic term is a "kinetic term" (a discrete version of a classical $\dot{v}^2$, and the rest is a "potential term". Expanding around the classical solution:
$$S[n + \delta n] = S[n] + \sum_{t,t'} \delta_{n_t} \frac{d^2S}{\partial_{n_t}\partial_{n_{t'}}} \delta_{n_{t'}}$$ , where
$$\frac{d^2S}{\partial_{n_t}\partial_{n_{t'}}} = P_{t,t'} = C^{-1}_{t,t'}$$ the propagator, or the inverse covariance, that can be calculated using numerical techniques.
Thus we run monte carlo simulations, collect a bunch of n(t) functions, and create the covariance matrix: $$C(t,t') = <\delta n_t \delta n_{t'}>$$.
If $n_t$ refers to s41(t), then the thing is quite simple, but now a new idea is that $n_t$ could be $n_t = s41(t) + s32(t) +s23(t) + s14(t)$. If at slice $t = 0$ we have $n_t$, then $n_{t+1}$ will be $n_{t+4}$, if one takes into account the extra structures too, thus $n_\tau = s41(t)$, $n_{\tau+1} = s32_{t}$, $n_{\tau+2} = s23_{t}$, where t refers to the integer slices,$\tau$ refers to the "fractional" slices. Practically, in a triangulation, if one takes the "slab" which is given by all simplices between two integer slices, that also defines a 3-dimensional sub-graph where the building blocks are tetrahedra and prisms related to s41/s14 and s32/s23 respectively. As it is a well-defined 3-dimensional structure it has an associated volume, thus it is a candidate for a new effective action with a new volume parameter.
Then the effective action written as $S[n_\tau]$ is now the function of the composite volume. Its second derivative $\partial n_\tau$ however is also a composite thing, as $n_\tau = n_0 + n_1 + n_2 + n_3$ and $n_{\tau'}$ is $n_4 + n_5 + n_6 + n_7$ (for example for t = 0, thus \tau = 0 , thus $\tau' = \tau+4\cdot 1$)
This leads to this composite derivative of the composite function.
There is no such thing. This is thoroughly hidden in the standard notation, but in order to take a partial derivative you must specify not only what you are varying but what you are keeping fixed. So $\frac{\partial}{\partial (x + y)}$ doesn't mean anything (just try to write down a definition!) until you specify what you are keeping fixed. Are you keeping $x, z$ fixed? Or $y, z$? Or $x - y, z$? Or something else? Different choices will give you different answers and you can verify this for yourself.