Rather than deal with performing a variety of operations involving many fractions, a fast approach is to plug some values into the original expression.

Let's see what happens when x = 2 and y = 4

We get: \(\frac{x^{-1} - y^{-1} }{ (xy)^{-1} (x-y)}=\frac{2^{-1} - 4^{-1} }{ (2\times4)^{-1} (2-4)}\)

Aside: \(2^{-1}=\frac{1}{2}=0.5\), \(4^{-1}=\frac{1}{4}=0.25\) and \(8^{-1}=\frac{1}{8}=0.125\)

So we get: \(=\frac{0.5 - 0.25 }{ (8)^{-1} (-2)}\)

\(=\frac{0.5 - 0.25 }{ (0.125)(-2)}\)

\(=\frac{0.25 }{ -0.25}=-1\)

Since answer choice A is the only answer choice that evaluates to be -1 when x = 2 and y = 4, the correct answer is A

Cheers,
Brent