So let us throw for a bit
with our non-constructive definition.
Maybe we can conclude something
about our tangent line from it and adjust it.
So how do we start it?
Firstly, let's start with that we have some function
f towards x and y and some given chosen point.
Assume that points a, b.
There is a tangent plane at this point.
We do not know how it was defined here,
what it is right in the equation
but let's just assume that the function
does have this tangent plane at this point.
I'm going to highlight this point.
Yeah, so what we're going to do,
we're going to consider a section of this plane and
the surface with one specific plane here.
The plane is y equal to b. What does it mean?
Basically, we are looking at
a screen that have been
placed on our three-dimensional graph,
at the point a, b parallel to that y axis.
So what we got as a result,
we got two things.
Firstly, we've got intersection of our square with plane,
abstention plane, and our surface as rolled
out of intersection with our surface.
We get function of the curve,
f to what's x and a fixed constant b here.
Two planes types are planned and our screen intersects,
by as you all know, a straight line.
So what do we get as the result?
We get the best approximation of plane,
tangent plane which approximates
all the surface and our curve afterwards,
x and b in particular.
Thus, the straight line in the intersection approximates
our curve f towards x and b perfectly.
So it's a tangent line, right?
So if it is tangent line,
then we can directly write
its slope which was previously called a derivative.
Now we are going to call it partial derivative.
So we're implying that there is
other variables for our function but we fixed
all that variables except
our variable x and thus we defined its slope.
Its change towards a change of
only one variable x at this given point.
Also, I'm going to start this out and set there's
two different notations for partial derivative,
in operator form with
nice partial sign here and as usual,
notation with a lower index x here in order to
state the variable to
which is a partial derivative was calculated.
So that's the definition of our partial derivatives.
Let us see how it works out on some examples.