I need to do the following Taylor expansion. Let's suppose we have a function $f(n,|\nabla n|^2)(\nabla (|\nabla n|^2),\nabla n)$, where by $(\nabla (|\nabla n|^2),\nabla n)$ I mean the dot product and $n$ is a real-valued three-variable function $n(\bf {r})=$ $n(x,y,z)$. I would like to find the first order Taylor expansion term of it around the point $n_0$ if the perturbation is $n_1$, so that $n=n_0+n_1$. I am doing it in two ways.
First way
I suppose, that the mathematically CORRECT is to treat $f(n,|\nabla n|^2)(\nabla (|\nabla n|^2),\nabla n)$ as a whole and calculate the derivatives of it considering that it is a product of two functions. But if I will try to do it in that way I will encounter with the derivatives like $\frac{d(\nabla (|\nabla n|^2),\nabla n)}{dn}$, which I don´t know how to calculate.
Second way
$f(n_0+n_1,|\nabla (n_0+n_1)|^2)(\nabla (|\nabla (n_0+n_1)|^2),\nabla (n_0+n_1))=[f(n_0,|\nabla n_0|^2)+(n_1+n_0-n_0)f_{n}|_0+(|\nabla n_0|^2+|\nabla n_1|^2+2(\nabla n_0,\nabla n_1)-|\nabla n_0|^2)f_{|\nabla n|^2}|_0][\nabla (|\nabla n_0|^2+|\nabla n_1|^2+2(\nabla n_0,\nabla n_1)), (\nabla n_0+\nabla n_1)]=[f(n_0,|\nabla n_0|^2)+n_1f_{n}|_0+2(\nabla n_0,\nabla n_1)f_{|\nabla n|^2}|_0][(\nabla(|\nabla n_0|^2)+2\nabla(\nabla n_0,\nabla n_1), \nabla n_0+\nabla n_1)]$
($f_{n}|_0$ means partial derivative with respect to $n$ at $n_0$ and $f_{|\nabla n|^2}|_0$ means partial derivative with respect to $|\nabla n|^2$ at $|\nabla n_0|^2$).
So the higher order terms, like $|\nabla n_1|^2$ and zero order terms, like $f(n_0,|\nabla n_0|^2) (\nabla(|\nabla n_0|^2),\nabla n_0)$ should not be considered and finally I have the following expression:
$f(n_0,|\nabla n_0|^2) (\nabla(|\nabla n_0|^2),\nabla n_1)+2f(n_0,|\nabla n_0|^2)(\nabla (\nabla n_0,\nabla n_1),\nabla n_0)+n_1f_{n}|_0(\nabla(|\nabla n_0|^2),\nabla n_0)+2(\nabla n_0,\nabla n_1)f_{|\nabla n|^2}|_0(\nabla(|\nabla n_0|^2),\nabla n_0)$
In this way, I am actually expanding $f(n,|\nabla n|^2)$ and $(\nabla (|\nabla n|^2),\nabla n)$ functions SEPARATELY.
So, could you please let me know, what is the problem with my calcs? And how the derivatives like $\frac{d(\nabla (|\nabla n|^2),\nabla n)}{dn}$ can be calculated.
Best Regards
Henrikh