of at exist and are continuous in a neighborhood of , then is differentiable at . When this happens, then in addition, the total derivative of is the linear transformation corresponding to the Jacobian matrix of partial derivatives at that point. The total derivative as a differential form When the function under consideration is real-valued, the total derivative can be recast using differential forms. For example, suppose that is a differentiable function of variables . The total derivative of at may be written in terms of its Jacobian matrix, which in this instance is a row matrix: The linear approximation property of the total derivative implies that if is a small vector (where the denotes transpose, so that this vector is a column vector), then Heuristically, this suggests that if are infinitesimal increments in the coordinate directions, then In fact, the notion of the infinitesimal, which is merely symbolic here, can be equipped with extensive mathematical structure. Techniques, such as the theory of differential forms, effectively give analytical and algebraic descriptions of objects like infinitesimal increments, . For instance, may be inscribed as a linear functional on the vector space . Evaluating at a vector in measures how much points in the th coordinate direction. The total derivative is a linear combination of linear functionals and hence is itself a linear functional. The evaluation measures how much points in the direction determined by at , and this direction is the gradient. This point of view makes the total derivative an instance of the exterior derivative. Suppose now that is a vector-valued function, that is, . In this case, the components of are real-valued functions, so they have associated differential forms . The total derivative amalgamates these forms into a single object and is therefore an instance of a vector-valued differential form. The chain rule for total derivatives The chain rule has a particularly elegant statement in terms of total derivatives. It says that, for two functions and , the total derivative of the composite function at satisfies If the total derivatives of and are identified with their Jacobian matrices, then the composite on the right-hand side is simply matrix multiplication. This is enormously useful in applications, as it makes it possible to account for essentially arbitrary dependencies among the arguments of a composite function. Example: Differentiation with direct dependencies Suppose that f is a function of two variables, x and