Linear Interpolation Error estimation

Suppose we have applied trilinear interpolation technique over grid points in the space $$A\times B \times C \times D={(x,y,z)|x\in A, y\in B,z \in C}$$} . The interpolant, which is piecewise function, is denoted by $$I(x,y,z)$$. My aim is to calulcate the error of interpolation, that is $$I(x,y,z)-f(x,y,z)$$. For a linear interpolation, as i know the error is calulated based on the formula $$I(x)-f(x)=\frac{(x_1-x_0)^2}{8}max|f''(x)|.$$ Even for this case I don't grasp how to estimate the error, since the function $$f(x)$$ is unknown, moreover it is needed that $$f(x)$$ has second order derivative. Please guide me to understand error esimation for linear interpolation and for tilinear interpolation, when functional forms $$f(x,y,z)$$ and $$f(x)$$ are unknown.