Nachdifferenzieren

Nachdifferenzieren is a German term that translates to "post-differentiate" or "differentiate afterwards." In mathematics, it refers to the process of differentiating a function after it has already been subjected to some other operation or transformation. This concept often arises in calculus and analysis when dealing with sequences of functions, integral equations, or other complex mathematical structures.

For example, one might encounter situations where a sequence of functions converges pointwise to a limit function. If the individual functions in the sequence are differentiable, the question arises whether the limit function is also differentiable and how its derivative relates to the derivatives of the sequence members. Nachdifferenzieren in this context would involve examining the derivative of the limit function.

Similarly, in solving integral equations, a method might involve finding a sequence of approximate solutions that converge to the exact solution. If these approximate solutions are differentiable, the process of finding the derivative of the exact solution might involve nachdifferenzieren. The conditions under which nachdifferenzieren is permissible are crucial and depend on theorems like Arzela-Ascoli theorem or uniform convergence of derivatives. These theorems provide criteria to ensure that the derivative of the limit function exists and can be obtained by differentiating the sequence members. The term is less commonly used in everyday mathematical discourse but appears in specialized literature concerning advanced calculus and analysis.