differential

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

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

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

$$\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}-f^{\prime }(x)=⟨\mathrm{\Delta }x⟩.$$

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

$\mathrm{\Delta }f=(f^{\prime }(x)+⟨\mathrm{\Delta }x⟩)\mathrm{\Delta }x=f^{\prime }(x)\mathrm{\Delta }x+⟨\mathrm{\Delta }x⟩\mathrm{\Delta }x.$ (1)

This result may be uttered as the

Theorem.  If the derivative $f^{\prime }(x)$ exists, then the increment $\mathrm{\Delta }f$ of the function corresponding to the increment of the argument from $x$ to $x+\mathrm{\Delta }x$ may be divided into two essentially different parts:
$1^{\circ }$. One part is proportional to the increment $\mathrm{\Delta }x$ 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 $⟨\mathrm{\Delta }x⟩\mathrm{\Delta }x$ to the increment $\mathrm{\Delta }x$ of the argument tends to 0 along with $\mathrm{\Delta }x$.

As well, the converse of the theorem is true.

By Leibniz, the former part $f^{\prime }(x)\mathrm{\Delta }x$ 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 $x+\mathrm{\Delta }x$.

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

$$dx=\mathrm{\hspace{0.33em}1}\cdot \mathrm{\Delta }x=\mathrm{\Delta }x.$$

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

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

It follows

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

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

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

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

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

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

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