Help with a custom coordinate transformation

I’m just starting to use Mathematica and can’t wrap my head around the task I have at hand. It comes from field-theory.

(Sorry I had to insert formulas as images instead of proper LaTex as it would complain I have some unformatted code)

I have a function which depends on other functions of coordinates and their first derivatives

This is my setup:

1. apply a custom coordinate transformation $(x^1,x^2,…) to (x’^1, x’^2, …)$. By custom I mean that it’s not necessarily between "standard" coordinate systems like "spherical to cartesian", etc — just some transformation law which I need to provide explicitly. More over, this law linearly depends on a set of parameters $u^1, u^2, …$

2. this transformation also induces a linear transformation on $A_1, A_2, …$ via some other custom law which I will also provide (not necessarily tensors!):

The end goal is to calculate the difference

up to first order in $u$, so terms denoted by $left{ldotsright}_k$ is my answer.

If I was to do it manually, this would be my plan:

1. Note that LHS in (2) is a function of $x’$ — I need to expand it in $x$ with the help of (1). Only terms up to the first order in $u$ are necessary! So after this expansion I expect LHS to contain $A_i'(x)$ and a combination of derivatives $partial A’_i/partial x’^k$ multiplied by $u$ with $Y^i_k$

2. Then I need to calculate all the derivatives $partial A’_i/ partial x’^k$ taking into account result from the previous step and expanding derivatives by $x’$ in terms of $x$ according to (1). Here, again, I only need to keep the first order in $u$

3. Next, I would put all this mess into the LHS of (3), where I would need to use Taylor expansion of $L$ to calculate the difference.

How can I automate this kind of calculations with Mathematica?