Prove Cauchy-like determinant formula

Question

How can you prove the following determinant formula, where the determinant is the same as that of the Cauchy matrix $$\det [ \frac{1}{1 - x_i y_j} ]_{i, j = 1}^n = \frac{\prod_{1 \leq i < j \leq n} (x_i - x_j)(y_i - y_j)}{\prod_{i, j = 1}^n (1 - x_i y_j)}.$$