The differential calculus arises from the study of the limit of a quotient, Δ *y* /Δ *x,* as the denominator Δ *x* approaches zero, where *x* and *y* are variables. *y* may be expressed as some function of *x,* or *f(x),* and Δ *y* and Δ *x* represent corresponding increments, or changes, in *y* and *x.* The limit of Δ *y* /Δ *x* is called the derivative of *y* with respect to *x* and is indicated by *dy* / *dx* or *D* _{ x } *y* :;e3;none;1;e3;;;block;;;;no;1;4224n;185422n;;;;;eq3;comptd;;center;stack;;;;;CE5The symbols *dy* and *dx* are called differentials (they are single symbols, not products), and the process of finding the derivative of *y* = *f(x)* is called differentiation. The derivative *dy* / *dx* = *df(x)* / *dx* is also denoted by *y′,* or *f′(x).* The derivative *f′(x)* is itself a function of *x* and may be differentiated, the result being termed the second derivative of *y* with respect to *x* and denoted by *y″, f″(x),* or *d* ^{2} *y* / *dx* 2. This process can be continued to yield a third derivative, a fourth derivative, and so on. In practice formulas have been developed for finding the derivatives of all commonly encountered functions. For example, if *y* = *x* n , then

Everyday calculations of velocity usually divide the distance traveled by the total time elapsed, yielding the average velocity. The derivative *f′* ( *t* ) = *ds* / *dt,* however, gives the velocity for any particular value of *t,* i.e., the instantaneous velocity. Geometrically, the derivative is interpreted as the slope of the line tangent to a curve at a point. If *y* = *f* ( *x* ) is a real-valued function of a real variable, the ratio Δ *y* /Δ *x* = ( *y* _{2} - *y* 1)/( *x* 2 - *x* 1) represents the slope of a straight line through the two points *P* ( *x* 1, *y* 1) and *Q* ( *x* 2, *y* 2) on the graph of the function. If *P* is taken closer to *Q,* then *x* 1 will approach *x* 2 and Δ *x* will approach zero. In the limit where Δ *x* approaches zero, the ratio becomes the derivative *dy* / *dx* = *f′* ( *x* ) and represents the slope of a line that touches the curve at the single point *Q,* i.e., the tangent line. This property of the derivative yields many applications for the calculus, e.g., in the design of optical mirrors and lenses and the determination of projectile paths.

- Introduction
- The Differential Calculus
- The Integral Calculus
- Bibliography

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.