Let x₀, x₁, …, x_n be equally spaced values, so that x_i = x₀ + ih, for 1≤ i ≤ n. Suppose that the values f₀,f₁, …,f_n are known, where f_i = f(x_i), for some function f. The first differences are defined, for 0 ≤ i<n, by Δ‎f_i = f_i + 1−f_i. The second differences are defined by Δ‎²f_i=Δ‎ f_i + 1−Δ‎ f_i and, in general, the k‐th differences are defined by Δ‎^kf_i=Δ‎^k−1f_i + 1−Δ‎^k−1 f_i. For a polynomial of degree n, the (n + 1)‐th differences are zero.

These finite differences may be displayed in a table, as in the following example. Alongside it is a numerical example.

With such tables it should be appreciated that if the values f₀, f₁, …, f_n are rounded values then increasingly serious errors result in the succeeding columns.

Numerical methods using finite differences have been extensively developed. They may be used for interpolation, as in the Gregory–Newton forward difference formula, for finding a polynomial that approximates to a given function, or for estimating derivatives from a table of values.