[SOUND] So let us proceed with some basic definitions further.
Firstly, let us can assume the following annotations first.
We start with the change of a function,
which is simply how much the value of function change from
point a to some point x, which is drawn on slide here.
And also the same thing applies to the argument,
here is the change of the argument.
We will call it delta f and delta x, respectively.
The thing that we are actually interested in is
the linear part of this function change.
It's normally called a differential or DF.
Basically the differential is your childhood hopes and dreams.
Because imagine that you were in your childhood at the point a.
And here we are assuming that you are a super linear,
it's the simplest function of all.
And then you are supposed that you were growing like a linear function and
life was the most happiest life of all.
And you've actually changed as a linear function by your differential.
Well, real life is not the simplest one.
So this differential is not necessarily coincides with the real function's change.
So in order to understand how they're connected,
we need to introduce ourselves to the concept of differentiable function.
Differentiable function is basically the things that we are expect
from any decent function here, where it's linear approximation,
linear understanding of change, is there a differential?
It's kind of the same as function change.
So we're actually mathematicians and
works approximately without any other continuation
is kind of bad thing and not exactly math here.
So we need to define what is approximately actually means.
In our case it means that we infinitesimal to the change of argument here.
In our annotation, It is written as or all from as functions argument.
Well, the single variable functions, as you may remember, is the simplest case.
Errors here so here being differentiable simply
means that we are just having a derivative.
In other words, differentiable functions are functions that can be closely fitted
by straight lines or piece by straight lines, which is kind of works, too.
Now we are going to move to our Magics and
try to calculate some basic derivatives by definition.
See you in the next more interesting video.