Detecting the correlation between function inputs

Question

Consider the function

$y = f(X) , f : [0,1]^n \rightarrow R $

1. What could be possible ways to detect the correlation among the inputs $$x_i, i \in \{1,2,...n \}, \text{ where } X = (x_1,x_2,..x_n) $$ w.r.t. their impacts to the output $y$?

2. How to form the covariance / correlation matrix using $f$, if possible?

I am interested in both analytical (e.g. using directional derivatives $\frac{ \partial y}{ \partial x_i} $) and numerical (e.g. using Monte Carlo) ways.