Unizor - Derivatives - Notation

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

Derivatives - Notation

We have introduced a concept of derivative at point x₀ of a real function f(x) defined in some neighborhood of this point as the following limit (if it exists):
lim_Δx→0[f(x₀+Δx)−f(x)]/Δx

Let's emphasize once more that this definition is valid only if the limit above exists.
For some functions, like f(x)=2x this limit exists at any point of their domain. For some others, like f(x)=|x|, there are points (in this case x=0) where this limit does not exist, while at others (x≠0) it does.

The limit defining a derivative of function f(x) at point x₀, if it exists, is some real value defined by both the function itself and a point x₀ in its domain. Therefore, for all these points of a domain of function f(x) where this limit exists the derivative is a new function defined at all these points.
This new function, a derivative of function f(x), 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 first derivative of function f(x).
The domain of the derivative f'(x) 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).

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:
Δf(x) = f(x+Δx)−f(x)
Then the expression of a derivative of function f(x) at point x would look like this:
f'(x) = lim_Δx→0Δf(x)/Δx
Using the symbol f' to denote a derivative is attributed to Italian mathematician Lagrange (1736-1813).

To make a notation even shorter, mathematicians have abbreviated the above expression based on limits even further:
f'(x) = df(x)/dx
or, sometimes, (d/dx)f(x).

In these cases expression dx is called a differential of argument x - an infinitesimal variable, and expression df(x) is called a differential of function f(x) - also an infinitesimal variable.
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 df and increment of its argument dx.

In particular, since it implies that these two infinitesimals are proportional to each other with a ratio f'(x) (a function of argument x), it makes sense to use the following notation that relates differentials with a derivative:
df(x) = f'(x)·dx

Newton (England, 1643-1727) used a dot above the function symbol to denote its derivative, which we will rarely use.
Euler used D_x f(x) as a notation of a derivative.

The process of deriving a derivative from a function is called differentiation. We will go through this process for major functions in the Examples.