rules for nonsmooth functions are very important in mathematics and mathematical physics since there are several important physical models whose behaviors are described by functions or functionals with a very limited degree of smoothness. The following chain rule is proved in the paper . Note all partial derivatives must be interpreted in a generalized sense, i.e., as generalized derivatives.Theorem. Let be a function of class (i.e. a continuous and differentiable function having continuous derivatives) and let be a function in with being an open subset of . Then and where is the mean value of the function at the point , defined as A more general chain rule formula for Lipschitz continuous functions has been found by Luigi Ambrosio and Gianni Dal Maso and is published in the paper . However, even this formula has very important direct consequences: using in place of , where is also a function and choosing , the preceding formula gives the Leibniz rule for functions This implies that the product of two functions of bounded variation is again a function of bounded variation, therefore is an algebra. BV(Ω) is a Banach algebra This property follows directly from the fact that is a Banach space and also an associative algebra: this implies that if and are Cauchy sequences of functions converging respectively to functions and in , then therefore the ordinary product of functions is continuous in with respect to each argument, making this function space a Banach algebra. Generalizations and extensions Weighted BV functions It is possible to generalize the above notion of total variation so that different variations are weighted differently. More precisely, let be any increasing function such that (the weight function) and let be a function from the interval ⊂ℝ taking values in a normed vector space . Then the -variation of over is defined as where, as usual, the supremum is taken over all finite partitions of the interval , i.e. all the finite sets of real numbers such that The original notion of variation considered above is the special case of -variation for which the weight function is the identity function: therefore an integrable function is said to be a weighted BV function (of weight ) if and only if its -variation is finite. The space is a topological vector space with respect to the norm where denotes the usual supremum norm of . Weighted BV functions were introduced