*Notes to a video lecture on http://www.unizor.com*

__Concept of Differential__

A concept of

**differential**of a smooth function

**at point**

*f(x)***was briefly introduced when we defined a**

*x=x*_{0}**derivative**of a function as a linear function of an infinitesimal increment of its argument with a coefficient of proportionality equal to a derivative of this function at point

**.**

*x=x*_{0}The notation

*d*

*f(x*_{0})/*d*, which was introduced when we defined a concept of derivative, reflects this definition.

**x=f**^{ I}(x_{0})Here

**is any point in the domain of function**

*x*_{0}**,**

*f(x)**d*is an infinitesimal increment of argument

**x****from this point and**

*x**d*is

**f(x**_{0})**differential**of function

**introduced above.**

*f(x)*The notation that uses

*d*instead of Δ implies that we are not talking about just any particular increment, but about a process of decreasing this increment to zero, thus making it an infinitesimal variable.

Using this type of notation, we can write the definition of a derivative as follows:

*[*

= lim

**f**=^{ I}(x_{0})= lim

_{dx→0}*]*

**f(x**d_{0}+**x)−f(x**_{0})*/d*

**x**This implies that

[

*]*

**f(x**d_{0}+**x)−f(x**_{0})*/d*=

**x**

**f**^{ I}(x_{0})+εwhere

*is another infinitesimal variable.*

**ε**Next transformation:

**f(x**d_{0}+**x)−f(x**

= fd_{0}) == f

^{ I}(x_{0})·**x+ε·**d**x**or, using the definition of "little

*" as infinitesimal of higher order,*

**o**Δ

*=*

**f(x**_{0})*Δ*

**f(x**_{0}+*=*

**x)−f(x**

= fd_{0}) == f

^{ I}(x_{0})·**x+o(**d**x)**=

*d*

**f(x**d_{0}) + o(**x)**Let me emphasize again that in the above statement

*d*is not just any increment of argument

**x***, but an infinitesimal variable representing an increment converging to zero.*

**x**Similarly,

*d*is an infinitesimal variable representing an infinitesimal function increment during the process of an increment of an argument converging to zero.

**f(x**_{0})From the definition of a differential

*d*

**f(x**d_{0})=f^{ I}(x_{0})·**x**we see that differential is a function of two arguments: a fixed point

*within a domain of function*

**x**_{0}*and an infinitesimal increment of an argument*

**f(x)***d*.

**x**Since

*is any fixed point, we can talk about a function differential at any value of argument*

**x**_{0}*and use the notation*

**x***d*:

**f(x)***d*

**f(x)=f**d^{ I}(x)·**x**Here is an illustration of a concept of a differential.

The blue line represents a function, red line - a tangential to it at point

**.**

*A*Segment

**represents an increment of the argument.**

*BD=AC*Segment

**is an increment of a function.**

*DE*Segment

**is function's differential - an increment of a value along the tangential line proportional to the increment of the argument.**

*DF*When point

**moves closer to point**

*C***, decreasing the increment of the argument, both segments**

*A***and**

*DE***decrease as well, while points**

*DF***and**

*E***are getting closer to each other, illustrating that increment of a function and its differential are infinitesimals if the increment of the argument is infinitesimal.**

*F*What's extremely important is that these two infinitesimals are of the same order as an increment of the argument, while difference between them, segment

**, is an infinitesimal of a higher order.**

*EF*Recall the

*Taylor series*for function

**with expansion center**

*f(x)***:**

*x*_{0}

*f(x)=*Σ_{n≥0}[*f*

^{ (n)}**(x**_{0})·(x−x_{0})^{n}/(n!)**]**

Let's set

**,**

*dx=x−x*_{0}Δ

*f(x*_{0})=f(x)−f(x_{0})and present this series as follows:

Δ

*f(x*

= f_{0}) == f

^{ I}(x_{0})·*d*+...

**x+f**d^{ II}(x_{0})·(**x)²/2**According to our definition of the differential, this can be expressed as

Δ

*f(x*_{0}) =*d*

**f(x**_{0}) + o(dx)which illustrates the same concept: increment of a function is of the same order as its differential and they differ by an infinitesimal of a higher order than increment of the argument.

It's appropriate to note here that the concept of a differential of a function justifies the Leibniz's notation for a derivative:

*f*^{ I}(x) =*d*

**f(x)/**d**x***Conclusion*

Differential

*d*of a function

**f(x**_{0})*at some fixed point*

**f(x)***of its argument is an infinitesimal variable proportional to infinitesimal increment of the argument*

**x**_{0}*d*with a coefficient of proportionality equaled to a derivative of this function at chosen point

**x***:*

**x**_{0}*d*

**f(x**d_{0}) = f^{ I}(x_{0})·**x**Differential differs from increment in a sense that increment is a fixed difference between two values, while differential is an infinitesimal variable.

Thus, Δ

*is a fixed number that is equal to a difference between the incremented value of argument*

**x***and its base value*

**x=x**_{1}*:*

**x=x**_{0}Δ

*=*

**x***−*

**x**_{1}

**x**_{0}But differential is an infinitesimal variable {

*−*

**x**_{1}*} in the process of*

**x**_{0}*→*

**x**_{1}*.*

**x**_{0}As soon as we switch from a fixed Δ

*to a process by considering Δ*

**x***, increment of an argument Δ*

**x→0***becomes its differential*

**x***d*.

**x**Similarly, Δ

*is a fixed number that is equal to a difference between the value of a function at incremented value of the argument and the value of a function at the base value of the argument:*

**f(x)**Δ

*=*

**f(x)***−*

**f(x**_{1})

**f(x**_{0})As

*→*

**x**_{1}*, function increment Δ*

**x**_{0}*is getting smaller and relative difference between its values and corresponding values of differential*

**f(x)***d*are getting smaller as well in a sense that

**f(x**_{0})*lim*[Δ

_{Δx→0}*]/*

**f(x)***d*

**f(x) = 1**because

*lim*[Δ

_{Δx→0}*]/Δ*

**f(x)**

**x = f**^{ I}(x)while

*d*

**f(x)/**d**x = f**^{ I}(x)and

*d*means the same as Δ

**x***, that is a process of infinitely decreasing increment of an argument.*

**x→0**
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