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Hi there.

Chapter six of the previous session We finished.

You will recall in chapter five partial derivatives and two

We saw the main definitions storey integral.

Chapter six in the derivative calculation methods We saw and did in some applications.

This calculation methods of these derivatives chain derivatives

The most basic element of a chain derivatives I have seen the concept of using the full derivatives,

I have seen the concept of directional derivatives and here we derive the concept of the gradient.

Now that the seventh chapter in this section more We will see further applications.

In many places these applications to business will work.

Also, in themselves purpose help

useful for producing additional topics issues.

On two main issues here We will stand, Taylor series and extreme value.

We mean the end of the minimum and maximum values find problems.

Or it is also deeper than the more modern The concept of finding the best method.

To find the best in the optimization we say.

Wherein finding the best optimization most

extreme values on the most common issues We will not.

With only the most basic local extreme values We'll see.

Once you understand the concepts here, the Our subject is very valuable

much water, the number of variables in functions

The second part of the more advanced applications

it is also about the other three applications We'll see.

Now let's start from the Taylor series.

I told them both on their own important objectives, as well

allowing other expansions as the vehicle Methods and applications.

I want to start with a remembered.

I hope that in a univariate function may remember for y is equal to x

Taylor series function given Is there a way going up.

This way, this way he follows the logic.

When we saw the call tangent The function reasonably

x is zero approach especially from the value of the function

and its derivatives using an approximately able to account values.

This too far from zero when x in better

values less than approximations We can find incorrect values.

After you do this to mind a question like this coming

I wonder, can we do it better said.

If you look here a constant value is there.

Has the value of the function value and derivative.

something already given x is zero.

So here's a force of first-degree There function.

I wonder if it is to expand n'yinc extent

What if we take as a power function We can think of.

To do this, add the hard numbers.

First-degree functions, quadratic power function and the degree n'yinc

terms up to take them as they collected hope to find a better solution.

It is about the degree with the function n'yinc

power function or the Taylor polynomial and we say x

the zero point of the function y equal to the value you get.

Yet this force at the point x is zero

The function of the derivative function Whether you are equal.

Bilaniha the floor of the second-order term number

It is the function we choose the second come to derivatives.

Here we see the following.

So here it is said already.

k'yınc degrees, the order derivative k'yınc

x is zero in the approximate function of the function given that

function of the actual total value is equal to we say.

And here when they made this equality As you can see each derivative

Although we have started buying from a n'yinci a terms of facing forward.

Here a future of power minus one.

When we take one more derivative of the times future minus 1.

As you can see in terms of a factorial Do you produce here and it K

the actual number of times given to us

order derivatives of the function of k'yınc

k'yınc to calculate the factorial We find from the division.

If we place them here in this f function zero first order derivatives

order derivatives of the second order derivative n'yinc In order derivatives are achieved.

It is a little more daring infinite polynomial If we're getting infinite series.

Of course, now we face when we get an infinite series

it will be your problem, the yakınsayıp convergence is that it

continue its beginning as a matter of while

the Taylor series, infinite series 're getting.

Yet it had a special case You will recall.

x is zero in many cases than zero to select provides an easy software.

For example, where x is reset to fall only x, the x's

frame, just as the cube of x simpler A software expenses.

This is what we call the Maclaurin series.

These are our single variable functions we see issues.

If y'all you forget it briefly here the main I have transferred the idea.

It's like opening a useful There is also indication:

Here we say that x x x minus x is zero to zero in

The difference between zero difference x in X, Delta'yl was showing.

The former first letter of the word differential d' Taken from the Greek.

If we see the delta with the letter x in the Delta x plus delta x where x would be zero.

But the x in x were zero.

So at a point x as delta x the difference

What happens when we give the function here 're getting expansion.

This is a useful initiative.

As it is a difference:where certain We're walking around an x zero.

Here's the function of any

variable expansion around the point offers.

We obtain the structure of these expansions.

Now we continue the same philosophy tangent

function of two variables in the right place There tangent plane.

We know that the tangent plane.

Y is zero, x is zero of gene functions the value at.

The value of derivatives, but one variable of functions of a

wherein when one derivatives have two derivatives It is natural.

It, s also reset x minus x minus y y Reset multiplications.

Were cut here in univariate There was only x minus x is zero.

Again when we come to the Taylor series

the same logic as in one variable are continuing.

Constant term in terms of x or x minus x first in terms of zero forces.

x minus x minus y is zero zero y in terms of Used forces.

Subject here is the square of x minus x is zero as the square of y minus y is zero there.

And still the second-degree term x

Reset minus x minus y is zero multiplied by y is there.

We need to take such a term.

Gene X, third-degree term of terms

cube, X, to the square of the terms of y'l first of terms

of the product of force and symmetrical, the x's

the square of the first force and y y cube.

And this is just one way when we open

as a variable function series expansion we see.

Similarly, we can make infinite series.

Convergence needs to be dealt with.

And as you can see in a very natural way opening.

I need to pay attention to this one.

Functions of one variable, for example, only There was one second derivatives.

However, here are three of the second derivative And second derivatives with respect to x and y

once as the second derivative of x y has the second derivative at a time.

And one thing to note here

two coefficients in front of these mixed derivatives is coming.

The reason for this is obvious.

We know from Clairaut's theorem.

There is a also a f x f y y x

and they are equal to each other if function if it provides the necessary continuity.

Therefore, this complex derivative mixed derivatives will come twice.

As an x and y a y x.

According to E means that they are equal in toward the front of the two coefficients.

Similarly, for example wherein E, X diced The term bi

time, otherwise you can not create the future third order derivative with respect to x, similar

According to the third order derivative of y, but second order with respect to x

derivatives, first order derivative of y by three You can create a kind.

First derivative with respect to x bi, bi more with respect to x derivatives, then the derivative with respect to y

like before you can get to x, then y, then again be able to get to x.

Again, similar to y ago, before xa

One more time then with respect to x you can get.

Therefore, this three times for coming here comes the factor of three.

This is the reason for this.

As you can see this same logic approaching the same

With a little generalized expansions gives us.

Yet this useful only as an illustration also gave variable functions.

Will place an x next to zero, y is zero x near the delta x delta y and y are they

If we think we've made incremental exactly wherein A series occurs.

The difference of these functions b alone derivatives x

variables x and y in y is not zero at zero is achieved.

This considerable flexibility in applications provides.

Expansion around any point gives.

Here, O, second order, third order numerical derivatives

Same here, though there are values has the function of derivative order.

This is just one variable functions as it is.

Yet this e here, to highlight similarities For bi

If we work hard in the summer than one variable, first-order derivative and the first

degree function, second-order derivatives and n'yinc order of quadratic functions

and, derivative and n'yinc forces, it is Taylor seris,

s are in the open in this way.

The coefficients've just explained.

Why twice, three times why nike come from?

These are nothing to do with Pascal's triangle numbers.

When you need them You can easily find the look.

Do you already logically disconnect.

How many points will come here to say for example x y you'll find two that immediately.

Because f x y and x y and y x as you can create.

This number we look for the next couple of X x Y front?

Yet before x x y x y x, y

x as x, and only being able to create You can see that you can create three times.

Then come three times.

We look at the fourth order, there is still by x

third derivative, b y the time derivatives etc. will be.

This supremely easy to understand structure.

Now we are here for a variable 've seen.

We saw two variables.

But three variables can be.

Even the one variable can be.

Although the function of three variables, for example, our x, y and

z 't due to the constant term b will be again.

Yet only one variable with respect to x There were first-order terms.

Two variables in both the x and y according to While first-order terms

where b is the same structure according to the terms in z will be.

Again we look to the second order derivatives only There are still only in variable f x x.

But we saw a little earlier bivariate when f x x

as well as have fun b y f x y have also.

But I came here for our bi of the z'l There terms.

You see, all of a sudden z'l terms volume increases.

In the past one to six, while three terms We arrive.

Because z y'yl also be mixed derivative,

also be mixed with X and their derivatives front

the same for the two time coefficients x

In the same manner as a case of twice again income.

Because f as f x z z x are also available.

For him, this dual income.

If we look at the third-order terms will increase as you can see.

But this is a very simple bit of logic two preceding terms

variable in terms of Pascal's triangle I told you here.

So the force of a plus b n'yinc from opening.

Wherein a plus b plus c is

coming from the opening forces n'yinc The term will be.

For example, third-ku, order and third here we look strong terms

x, y and x in all the terms in z'l If you receive future bi times.

But this one bi, bi grain it, it bi one

If you get that you think f x y z

When e, it in a different order

for the future of this gene in different orders terms will occur wherein

According to him, the coefficients on the front and will increase.

This behold it in order twentieth We do not take the benefits.

Its structure which is important, the logic understand.

In the examples we have already done a little more to reinforce these concepts.

n-variable function, so that we look to Blend until smooth

There is a structure where the variables now x y z course, he is not able to continue.

x A x two x three x j x j plus one x n as we can go.

Yet they will be in constant terms.

Will be the first order terms.

There will be a second order derivatives.

Will be the third order derivatives.

Here again, as in gösterince with coefficients the same structure constant terms

some first order derivatives and first-ku,

including terms of degree of x again with the second moment

The combination of second-order terms, and this will be the way to go.

Coefficients in front of the still earlier j before logic, the jth

After variable by variable k'yınc When you take the derivative of this variable k'yınc

and j'yinc variable in reverse order so derivatives thereof is equal to

The second term longitudinal front two

coefficients of three coefficients, and so on is the future.