  A peculiar feature of It\^o's calculus is that it is an integral calculus
that gives no explicit derivative with a systematic differentiation theory
counterpart, as in elementary calculus. So, can we define a pathwise stochastic
derivative of semimartingales with respect to Brownian motion that leads to a
differentiation theory counterpart to It\^o's integral calculus? From It\^o's
definition of his integral, such a derivative must be based on the quadratic
covariation process. We give such a derivative in this note and we show that it
leads to a fundamental theorem of stochastic calculus, a generalized stochastic
chain rule that includes the case of convex functions acting on continuous
semimartingales, and the stochastic mean value and Rolle's theorems. In
addition, it interacts with basic algebraic operations on semimartingales
similarly to the way the deterministic derivative does on deterministic
functions, making it natural for computations. Such a differentiation theory
leads to many interesting applications some of which we address in an upcoming
article.
