An advanced sensor detects nanoscale gaps using inductance. Consider an LC circuit with a parallel-plate capacitor having plate area A and known inductance L. The charge oscillates at a frequency f when the capacitor plates are separated by a distance x. The relationship between 𝑥, 𝜖₀, 𝐴, 𝑓, and 𝐿 is given by: x=4π²Aϵ₀f²L.

When the plate separation changes by a small amount ∆x, the oscillation frequency changes by ∆f. Using the given relationship, derive an expression approximating ∆x/x.