From Arnold's “Old and recent stories”:

The textbook of Zel'dovich (physicist) defines derivative as `the difference quotient, where difference of the argument is sufficiently small.'

He did not want to consider any limits, because in his words, `there is no sense to consider the difference smaller than $10^{-10}$: the structure of the space and time in such closeness is not described by the mathematical continuum.'

-- We are, - he said - always interested in the ratio of the finite differences, and the derivatives of mathematicians - it's just approximate mathematical formulas to calculate the ratios of finite differences.