If a real function f has the derivativePlanetmathPlanetmath at a value x of its argumentMathworldPlanetmathPlanetmath, then the absolute valueMathworldPlanetmathPlanetmath of the expression


may be made smaller than any given positive number by making |Δx| sufficiently small.  If we generally denote by Δx an expression having such a property, we can write


This allows us to express the increment of the functionMathworldPlanetmathf(x+Δx)-f(x):=Δf  in the form

Δf=(f(x)+Δx)Δx=f(x)Δx+ΔxΔx. (1)

This result may be uttered as the

Theorem.  If the derivative f(x) exists, then the increment Δf of the function corresponding to the increment of the argument from x to x+Δx may be divided into two essentially different parts:
1. One part is proportional to the increment Δx of the argument, i.e. it equals this increment multiplied by a coefficient f(x) which is on the increment.
2. The ratio of the other part ΔxΔx to the increment Δx of the argument tends to 0 along with Δx.

As well, the converseMathworldPlanetmath of the theorem is true.

By Leibniz, the former part f(x)Δx is called the differentialMathworldPlanetmath, or the differential increment of the function, and denoted by df(x), briefly df.

It is easily checked that when one has set the tangent lineMathworldPlanetmath of the curve at the point  (x,f(x)),  the differential increment df(x) geometrically means the increment of the ordinate of the corresponding to the transition from the abscissa x to the ascissa x+Δx.

The differential of the identity functionMathworldPlanetmath (f(x)x,  f(x)1) is

dx= 1Δx=Δx.

Accordingly, one can without discrepancies denote the increment Δx of the variable x by dx.  Therefore, the differential of a function f gets the notation

df(x)=f(x)dx. (2)

It follows

f(x)=df(x)dx, (3)

in which the differential quotient is often replaced using a “differentiationMathworldPlanetmath operator”:

f(x)=ddxf(x) (4)

Remark.  One can write certain for forming differentials.  For example, if f and g are two differentiable functions, one has

d(f+g)=df+dg,d(fg)=fdg+gdf. (5)

Naturally, they seem trivial consequences of the sum ruleMathworldPlanetmath and the product ruleMathworldPlanetmath, but they include a deeper contents in the case where f and g depend on more than one variable (see total differential (http://planetmath.org/TotalDifferential)).
As for a composite function, e.g. h=fu, the chain ruleMathworldPlanetmath and (2) yield


i.e. simply



  • 1 Ernst Lindelöf: Johdatus korkeampaan analyysiin. Fourth edition. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
  • 2 E. Lindelöf: Einführung in die höhere AnalysisMathworldPlanetmath. Nach der ersten schwedischen und zweiten finnischen Auflage auf deutsch herausgegeben von E. Ullrich. Teubner, Leipzig (1934).
Title differential
Canonical name Differential
Date of creation 2014-02-23 15:34:46
Last modified on 2014-02-23 15:34:46
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 23
Author pahio (2872)
Entry type Definition
Classification msc 53A04
Classification msc 26A06
Classification msc 26-03
Classification msc 01A45
Synonym differential increment
Related topic Derivative2
Related topic LeibnizNotation
Related topic ExactDifferentialEquation
Related topic ProductAndQuotientOfFunctionsSum
Related topic TotalDifferential
Defines differential quotient