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

__Derivatives - Notation__

We have introduced a concept of

*derivative*at point

**of a real function**

*x*_{0}**defined in some**

*f(x)**neighborhood*of this point as the following limit (if it exists):

*lim*_{Δx→0}

**[**Δ

*f(x*_{0}+**Δ**

*x)−f(x)*]*/*

*x*Let's emphasize once more that this definition is valid only if the limit above exists.

For some functions, like

**this limit exists at any point of their domain. For some others, like**

*f(x)=2x***, there are points (in this case**

*f(x)=*|*x*|**) where this limit does not exist, while at others (**

*x=0***) it does.**

*x≠0*The limit defining a derivative of function

**at point**

*f(x)***, if it exists, is some real value defined by both the function itself and a point**

*x*_{0}**in its domain. Therefore, for all these points of a domain of function**

*x*_{0}**where this limit exists the derivative is a new function defined at all these points.**

*f(x)*This new function, a derivative of function

**, defined at all points of a domain of function**

*f(x)***where the above limit exists, is traditionally denoted as**

*f(x)***and called the**

*f'(x)**first derivative*of function

**.**

*f(x)*The domain of the derivative

**is a subset of a domain of function**

*f'(x)***since, in theory, the above limit might not exist at all points of domain of**

*f(x)***. If this limit exists at each point of domain of**

*f(x)***, the domain of a derivative**

*f(x)***coincides with the domain of original function**

*f'(x)***.**

*f(x)*The definition of a derivative as a limit of ratio of two infinitesimals, increment of function and increment of argument at some point, prompts us to use the following notation:

Δ

**Δ**

*f(x) = f(x+*

*x)−f(x)*Then the expression of a derivative of function

**at point**

*f(x)***would look like this:**

*x*

*f'(x) = lim*_{Δx→0}

*Δ*

**Δ**

*f(x)/*

*x*Using the symbol

**to denote a derivative is attributed to Italian mathematician Lagrange (1736-1813).**

*f'*To make a notation even shorter, mathematicians have abbreviated the above expression based on limits even further:

*f'(x) =**d*

**f(x)/**d**x**or, sometimes,

*(d/dx)*.

**f(x)**In these cases expression

*d*is called a

**x***differential*of argument

**- an infinitesimal variable, and expression**

*x**d*is called a

**f(x)***differential*of function

**- also an infinitesimal variable.**

*f(x)*These notations are attributed to German mathematician Leibniz (1646-1716). They are as popular as the Lagrange's notation above. They have certain advantage since they remind us that a derivative is a ratio of two infinitesimals - increment of function

*d*and increment of its argument

**f***dx*.

In particular, since it implies that these two infinitesimals are proportional to each other with a ratio

**(a function of argument**

*f'(x)***), it makes sense to use the following notation that relates**

*x**differentials*with a

*derivative*:

*d*

**f(x) = f'(x)·**d**x**Newton (England, 1643-1727) used a dot above the function symbol to denote its derivative, which we will rarely use.

Euler used

*D*as a notation of a derivative.

_{x }**f(x)**The process of deriving a derivative from a function is called

*differentiation*. We will go through this process for major functions in the Examples.

## No comments:

Post a Comment