For a little more than sixty years the classical Ito calculus existed only as an integral calculus without a definition of a stochastic derivative of a semimartingale with respect to another semimartingale that's an anti Ito-integral and that leads to a corresponding systematic theory of pathwise differentiation.  In 2004 I completed the elements of this theory (including a fundamental theorem of stochastic calculus, a chain rule, a mean-value theorem, other differentiation rules and formulas, and more), and parts of this work are published in my "A Differentiation Theory for Ito's Calculus" 2006 paper.