Total Derivative

In the mathematical field of differential calculus, the term total derivative has a number of closely related meanings.

• The total derivative (full derivative) of a function, of several variables, e.g., etc., with respect to one of its input variables, e.g., is different from its partial derivative . Calculation of the total derivative of with respect to does not assume that the other arguments are constant while varies; instead, it allows the other arguments to depend on . The total derivative adds in these indirect dependencies to find the overall dependency of on . For example, the total derivative of with respect to is
  Which simplifies to
  Consider multiplying both sides of the equation by the differential :
  The result will be the differential change in the function . Because depends on, some of that change will be due to the partial derivative of with respect to . However, some of that change will also be due to the partial derivatives of with respect to the variables and . So, the differential is applied to the total derivatives of and to find differentials and, which can then be used to find the contribution to .

• It refers to a differential operator such as
  which computes the total derivative of a function (with respect to x in this case).

• It refers to the (total) differential df of a function, either in the traditional language of infinitesimals or the modern language of differential forms.

• A differential of the form
  is called a total differential or an exact differential if it is the differential of a function. Again this can be interpreted infinitesimally, or by using differential forms and the exterior derivative.

• It is another name for the derivative as a linear map, i.e., if f is a differentiable function from Rn to Rm, then the (total) derivative (or differential) of f at x∈Rn is the linear map from Rn to Rm whose matrix is the Jacobian matrix of f at x.

• It is a synonym for the gradient, which is essentially the derivative of a function from Rn to R.

• It is sometimes used as a synonym for the material derivative, in fluid mechanics.