# derivative

A function (black) and a tangent (red). The derivative at the point is the slope of the tangent.

A derivative is the result of differentiating a function; that is, the infinitesimal change in a function caused by an infinitesimal change in the variable(s) upon which it depends. The derivative gives the rate of change of a function (the slope of its curve) at a particular point. Second and third derivatives give the rate at which the rate of change is changing and the rate at which the rate of rate of change is changing, respectively.

The derivative of a function *f(x)*, at the point *x*, is
defined to be the value of the ratio

[*f(x* + *h) - f(x)*]/*h*,

In the limit where the number *h* approaches
zero, and is denoted d*f* /d*x* or *f *'*(x)*.

Derivatives provide a means of describing many processes which can be viewed
as changing continuously in time. For instance, if the position of an object
at a time, *t*, is given by *p(t)* then the velocity (which is defined as the *instantaneous* rate of change of position),
obtained by differentiation, is
given by the derivative d*p*/d*t*.

In an article in 1996, Hugo Rossi wrote: "In the fall of 1972 President
Nixon announced that the rate of increase of inflation was decreasing. This
was the first time a sitting president used the third derivative to advance
his case for reelection."^{1}

## Partial derivative

The partial derivative is the ordinary derivative of a function
of two or more variables with respect to one of the variables, the others
being considered constants. If the variables are *x* and *y*, the
partial derivatives of *f*(*x*, *y*) are written *δf*/*δx* and *δf*/*δy*, or *D*_{x}*f* and *D*_{y}*f*, or *f*_{x} and *f*_{y}. The partial derivative of a variable with
respect to time is known as the **local derivative**.

### Reference

1. Rossi. Hugo. "Mathematics Is an Edifice, Not a Toolbox." *Notices
of the AMS*, 43 (10) (Oct. 1996).