Total Differentiation of implicitly defined function with many variables

I need help sorting out some basic calculus. I need to take the total derivate of the following function with respect to the variable . Note, the variables are y and where goes from 1 to 7, while all other symbols are constants:

I want to calculate the total derivative of with respect to the variable

Where satisfy the following set of equation seven equations in eight variables and . Assuming these equations have a solution and it is unique, what is happening is that I get to choose and for every choice of we get some values for . The total derivative I need to calculate is the first order condition I need to figure out an optimal choice of . It is easy to verify in the context of the problem that my solution is not a boundary solution. So anyway, satisfy the following set of equation seven equations in eight variables and . :

This is how I have solved this:

Need to compute:

I will take the total derivative of the constraints to solve for , ,

So, we get the following equations:

These equations are linear in , , . I can solve for them and plug back into ($\ref{eq:main}$) to get the final answer.

Is this correct? or am I doing something wrong? If this is correct, I'd appreciate it if you can recommend some reading I could read which covers implicit differentiation in more than two variables so I can understand how to think about this without having doubts. If this is incorrect, then could you please tell me why and then again recommend some place I can read about this. Thanks