Mathematical Analysis

Differential Calculus

What is the derivative of a function?

One of the most important, core concepts in modern mathematics and calculus is the derivative of a function—or a function derived from another function. A derivative is also expressed as the limit of Δy / Δx, also said as “the derivative of y with respect to x.” It is actually the rate of change (or slope on a graph) of the original function; the derivative represents an infinitesimal change in the function with respect to the parameters contained within the function.

In particular, the process of finding the derivative of the function y = f(x) is called differentiation. The derivative is most frequently written as dy / dx; it is also expressed in various other ways, including f'(x) (said as the derivative of a function f with respect to x), y’, Df(x), df(x), or Dxy. It is important to note that the differentials, written as dy and dx, represent singular symbols and not the products of the two symbols. Not all derivatives exist for all values of a function; the sharp corner of a graph, in which there is no definite slope—and thus no derivative—is an example.


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