## 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 Δ*y* / Δ*x*, 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 *D _{x}y.* 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.