In the following notation this is how we determine the partial deriatives of a function using the Implicit Function Theorem, but I haven't seen this notation before, and is a bit unsure of how exactly to read this, specifically the right side of the equation: $\cfrac{\partial \phi_i}{\partial x_j}=\cfrac{\cfrac{\partial (F_{(1)},F_{(2)},\ldots,F_{(n)})}{\partial ( y_1,\ldots,x_j,\ldots,y_n)}}{\cfrac{\partial (F_{(1)},F_{(2)},\ldots,F_{(n)})}{(y_1,\ldots,y_i,\ldots,y_n)}}$
Here is an image providing context for the entire thing, but it is really only the notation on the right side of the previously shown equation that I haven't seen before. Context for entire notation of IFT
The expression $${\partial(F_1,F_2,\ldots,F_n)\over\partial (y_1,y_2\ldots,x_j,\ldots,y_n)}$$ is meant to be the determinant of the matrix $$\left[\matrix{ {\partial F_1\over\partial y_1}&{\partial F_1\over\partial y_2}&\cdots& {\partial F_1\over\partial x_j}&\cdots &{\partial F_1\over\partial y_n}\cr {\partial F_2\over\partial y_1}&{\partial F_2\over\partial y_2}&\cdots& {\partial F_2\over\partial x_j}&\cdots &{\partial F_2\over\partial y_n}\cr \vdots\cr {\partial F_n\over\partial y_1}&{\partial F_n\over\partial y_2}&\cdots& {\partial F_n\over\partial x_j}&\cdots &{\partial F_n\over\partial y_n}\cr}\right]\ ,$$ and similarly for the expression in the denominator of the quoted formula. Note that a minus sign is missing.