# differential

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

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 $\Delta x$.

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

 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