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So now we know what is our tangent line supposed to look like, if it is exists.
But still we don't stated how should one decide whether
this function can be filtered by our tangent plane or not.
That's basically the concept of differentiability from our single-variate
case, but right now it is two variable case, or multi-variable case in general.
So when discussing this fittable concepts here,
we are going to fit our function with a plane or hyperplane, respectively.
So what we're going to do, we're going to just write with the very basics in here.
Our function which is f(x,
y) = our tangent line as we defined in the previous
video plus some kind of error, right.
This is the error that we get while we are approximating our function with
the tangent plane.
But what's that error?
As we previously remember in single-variate case,
this was little o, infinitesimal towards their change of our variable.
But right now we are actually looking at the case where have two variables,
and we were looking at some point (a, b) and point (x, y) at the plane.
So the change is actually, well quite a lot of things.
But we're going to refer to it as just a simple distance
between two points in the real plane.
And thus, by the Pythagoras Theorem, it's quite easily calculatable, right?
It's square root from sum of squares of,
The segments which are just sides of this triangle, so that's our definition.
Our function is differentiable if it is actually can be
approximated by a tangent plane as we stated as we defined in the video before.
So let us just make this point crystal clear for you.
Differentiable has a derivative.
That's right, but only for single-variate functions, okay.
For multivariate functions, differentiable is much more
powerful than the existence of the derivative itself.
And it's crucial here because as you can see, what actually changed?
The thing that has changed is the number of degrees of freedom,
the number of variables, right.
There were one variable, one degree of freedom.
And now it's two or more degrees of freedom.
That's if for only x, it's a good fit for only one variable,
its tangent plane is quite a good approximation.
It does not mean that for all the variables x and y in composite,
it's going to be the best thing that we can come up with.
So we need to carefully refer to this variable definition,
and do not feel comfortable just by calculating our partial
derivatives as it was in our single-variate case.
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