# differential

If a real function $f$ has the derivative at a value $x$ of its argument, then the absolute value of the expression

 $\frac{f(x\!+\!\Delta x)\!-\!f(x)}{\Delta x}-f^{\prime}(x)$

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

 $\frac{f(x\!+\!\Delta x)\!-\!f(x)}{\Delta x}-f^{\prime}(x)\;=\;\langle\Delta x\rangle.$

This allows us to express the increment of the function$f(x\!+\!\Delta x)\!-\!f(x):=\Delta f$  in the form

 $\displaystyle\Delta f\;=\;(f^{\prime}(x)\!+\!\langle\Delta x\rangle)\Delta x\;% =\;f^{\prime}(x)\Delta x+\langle\Delta x\rangle\Delta x.$ (1)

This result may be uttered as the

If the derivative $f^{\prime}(x)$ exists, then the increment $\Delta f$ of the function corresponding to the increment of the argument from $x$ to $x\!+\!\Delta x$ may be divided into two essentially different parts:
$1^{\circ}$. One part is proportional to the increment $\Delta x$ of the argument, i.e. it equals this increment multiplied by a coefficient $f^{\prime}(x)$ which is on the increment.
$2^{\circ}$. The ratio of the other part $\langle\Delta x\rangle\Delta x$ to the increment $\Delta x$ of the argument tends to 0 along with $\Delta x$.

As well, the converse of the theorem is true.

By Leibniz, the former part $f^{\prime}(x)\Delta x$ is called the differential, 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 line 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 $x\!+\!\Delta x$.

The differential of the identity function ($f(x)\equiv x$,  $f^{\prime}(x)\equiv 1$) is

 $dx\;=\;1\cdot\Delta x\;=\;\Delta x.$

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

 $\displaystyle df(x)\;=\;f^{\prime}(x)\,dx.$ (2)

It follows

 $\displaystyle f^{\prime}(x)\;=\;\frac{df(x)}{dx},$ (3)

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

 $\displaystyle f^{\prime}(x)\;=\;\frac{d}{dx}f(x)$ (4)

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

 $\displaystyle d(f\!+\!g)\;=\;df\!+\!dg,\;\qquad d(fg)\;=\;fdg+g\,df.$ (5)

Naturally, they seem trivial consequences of the sum rule and the product rule, 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. $h=f\!\circ\!u$, the chain rule and (2) yield

 $df(u(x))\;=\;f^{\prime}(u(x))u^{\prime}(x)dx\;=\;f^{\prime}(u(x))du(x),$

i.e. simply

 $dh\;=\;f^{\prime}(u)du.$

## References

• 1 Ernst Lindelöf: Johdatus korkeampaan analyysiin. Fourth edition. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
• 2 E. Lindelöf: Einführung in die höhere Analysis. 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