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Hello.

Hopefully last week's

As the information in a past session Did you revise.

Derivatives of functions of one variable in Do you remember the concept well.

But here, once again ad nauseam to you

We will see even at the expense of one variable the derivative of the function.

Our goal here initially for two

to generalize the derivative variable functions concept.

Partial derivatives of functions of several variables It's called derivatives.

Because for both x and y to which is If vary

According to him, it depends partly work, partly it depends.

A term derived from it.

The essence of the approach, I've said before variables respectively

that will temporarily freeze before y We will freeze,

hence function of only x function will be.

Then a function of one variable reduction would have.

This is the derivative of functions of one variable

entirely in a single variable functions as such.

We saw these at some point derivative We have seen an x near zero.

We choose a point x.

By combining these two, these two points combining

we obtain a beam with a right combine.

That change in the y direction of the slope divided in the x direction

In exchange, and x, x goes to zero it

at this point when receiving derivative derivative 're getting.

The two-variable functions Supremely easy generalization.

We say we x, y'l the

If we change both in function there will be confusion.

Let's change for him one by one.

What to replace one at a time should we

y is zero derivative at point x is zero We want to calculate.

So I'm dondural y ago.

See whether y wherein Let's think.

This year we are equating to zero.

X is exactly as you see above

The function of the function x is zero We're taking.

X minus x is zero in the denominator completely.

Be desired that it limits derivative obtained by We are.

As you can see in one variable functions such as derivatives.

As we say that x to navigation

was made according to the variable x, here x We're putting display.

And this year we x is zero at zero point We call this number to calculate.

Here x is zero y putting a zero

p is zero, or at the point of zero We calculated the call.

X is zero at this time x the same procedure We provide choosing.

See if we consider that there are x wherein y has a function.

This function of y in the y is zero would remove the value.

This change in the y direction.

And brought it to the change in the controlling stake.

As shown here, instead of where x Think of years.

This also happens at the limit of derivatives.

This is a slope finally.

To distinguish it from the other of y variable is

We use this symbol and it shows x is zero

When y is zero this limit calculated x zero y

calculated at zero to indicate that the We put a zero indicator.

How a number of these results

A number of derivatives in one variable at the point If giving.

However, functions of one variable from the Let's start.

Obtained wherein the derivative function We would.

We were obtained as follows; selecting the variables x Another point next to the variable x

choosing him away from the delta x the value of the function at the point

finding that the difference in the y direction in the vertical the difference was giving.

That when we divide the delta x were finding the slope of the function.

These beams were finding the slope.

This slope when receiving limit of x point

were finding the slope geometry as x We were at the point we find the rate of change.

This x for any x, which can

is a function for a variable x gave us.

Function of two variables in the same operation we can do.

Wherein a change in the y

We not only offer only a change in x, As we are using the delta x.

See also the year that the If you think

Also, we obtain the above terms.

The value of x is x plus delta function, The value of x is negative and y is kept constant.

This means that in the vertical, z change in giving.

That when we divide the delta x We find the slope of the curve.

At the limit of this derivative is happening.

As we show in this, again, this xa Accordingly, x is variable, with respect to x

derivatives informed him that we have received, because y We have temporarily freeze here, change

We have not y, the derivatives with respect to x is we say.

But here we show that as the df dx

So here we have used is in Greek, In Ancient Greek

d'y delta, delta small letters We use curved d are showing.

This is important for these two variants of each other To reserve.

In addition, the concept of a future full-derivatives

We will meet with the second of two variables of functions.

Here we find the partial derivatives.

D I'll show him, for her d is delta, or just one

Above shows a difference in essence says.

We can do the same process for years.

We are temporarily freezes the x, y We give a change.

So this zoom function x

z is frozen in time by the change in y We find change.

This gives the change in vertical again, this is changes in landscape.

This ratio gives an incline again.

According to the partial derivative of y we call it.

D. Again this curved, Old We use the Greek letter delta.

That's why we're using it, because that's for such is the way of universal representation.

However, in univariate function

As a point of the partial derivatives we'd like to calculate, i.e.

partial at a point on the previous page We have shown that derivatives.

f x above zero, f y above zero.

Where f x above zero this partial

x is zero at zero of the derivative function y We find value.

As a demonstration there, they equivalent to each other notations.

Sometimes it would be more convenient for short to write.

Sometimes it reviews a sense of meaning, which can be written to make the difference.

This partial derivative in a certain way again 're getting.

x on y is zero zero are calculated.

Compared to what you're getting, you're getting with respect to x.

Here, according to the partial derivative of y 're getting.

X is zero and y is zero at the point we calculate said.

Hence the derivative is a derivative of

connected to each other in this way function We're going.

Let's do a few examples, our example simple My pick of the functions.

Get a little more complicated, but they There is no difficulty.

Our main concept means that such a given function x plus two years.

means the partial derivative with respect to x, y fixed Keeping you mean.

Take the derivative with respect to x by x would be the one, y

taken to be constant when the derivative with respect to x zero.

In another interpretation of the x and y from one another is independent of y with respect to x

derivative is zero overlap as definitions, as an expression of independence.

When we take the derivative with respect to y is similar to

This time as a hard task x will do.

When we take the derivative with respect to y is a constant

y is derived based on the zero, so does not seem right here.

Two derivative of y to y gives the two.

Let's take a slightly different as a function.

Although there are still a simple multiplication x y.

This partial derivative with respect to x at y fixed is going on,

in front of it as though 3, 5, 17, as if there were.

Derivatives of these constants have not changed During the derivative of x

just arrive, then 1, y as in remains.

Similarly, the partial derivatives to y so that

x doing a hard task, he is staying.

derivative of y to y is from 1 to only x remains.

To see a slightly different direction x square + square'll pick two years.

This is now the partial derivative with respect to x We learned 2x coming here,

y is a constant which serves as its a does not contribute.

Similarly, the partial derivatives at x to y a

Because hard acts to y derivative does not make a contribution.

Because derivative 2y y frames, beginning 2, we find that the four.

Review of the fourth sine function we take to the force of the y'inc

y'inc to force the exponential can say.

Again, if we take the derivative with respect to x to y via

doing a hard task, where e to y remains.

4 According to the fourth power of the sine of x This 4 forward derivative

falls to the fourth power of a sine running out of 3, but the interior

sine derivative of x with respect to x to the requirements There, he also gives the cosine of x.

At first I said this te preparation information

of functions of one variable derivative chain rule.

Sinus 4 u If you say sine x, sine x 4, u will be the fourth force.

So would a composite function.

Taking the derivative of this with respect to x of the u 4 ua

we take the derivative, of u a with respect to x We take the derivative.

Here's derivative with respect to u 4 u 4 times the cube happens.

Then u also were in the sinus x A, the cosine gives the derivative with respect to x.

Telling them for a long time because

In my experience so far, students this work

or it can be a bit forgotten

going a little rusty in the practice of business information.

Revive them brighter I think the opportunity to bring.

Maybe a little boring if you know may

but it does not hurt to repeat it to everyone helpful.

This time when the derivative of y by x

fixed, so the fourth sinus x where the force is coming.

e to again derivative of y to y e years later.

You'll recall the exponential function is very almost a miraculous functions.

It is the derivative of a function to itself equal, it is equal to the integral.

E to thereby derivative of y to y In later years to give.

In this opportunity we are also remembered.

Visually the meaning of partial derivatives Let's review.

means that y is equal to a constant, say y = y0, y is the vertical plane passing through.

I have seen the equation of the plane.

x, y, z, all with first force it seemed.

But x and y can have a mind does not exist.

Then y = y0, y0 passing through a vertical plane is going.

This event middle figure z = f (x, y) function

perspective representation and the vertical the perspective of the plane representation.

Similarly, at x = x0, x0 namely y parallel to the axis

directly passing the vertical plane, these Built right on the vertical plane.

Now the single variable functions We know, two functions

If you take more than one variable each must provide in both.

Thus both geometrically geometrical

must be present on, so is a cross-section.

Y = z means y0'l = f (x, y) together When we received an intersection found

we're seen here that y = y0, x variable, which was built on this plane

16:08

Other functions x0 to x when we give with z

a function of y, the function

y0 is the partial derivative of the slope again, f (y).

This means that the x, y direction unit When you go to the curve

What we're up to on a tangent to it We're on the measure.

The slope of the tangent is on we see.

As in the univariate function very

variable functions and partial order partial derivatives can be calculated.

So when we calculate how the df dx take a

If we take the derivative and it is once again in square f

We called and d x squared, where the partial in derivatives

df dx one more partial with respect to x We can take the derivative.

Because for partial derivatives, that X and function y.

It is the same one variable in function f d d squared

We are writing, but we use x squared d's that the partial derivatives

to remind the curved d's, Old The Greek letter delta.

df dx as we take the derivative with respect to x We can take the derivative of y by.

This again is consistent with this article dy dx frames are writing for.

Y out if you pay attention, here outside, the inside x.

We call this complex mixed derivatives, both the x to y.

However, these two time derivative with respect to x taken.

According to y for x we process we can do.

to y dy'n again derivative dF can get.

This is f d y square frame gives the same d As in the square f d x squared.

But still, the partial derivative d's We are using.

Df dy'n this time the same process with respect to x By differentiating we can.

It looks like just above the button.

This mixed derivatives, i.e., y with X involving derivatives.

But as different.

Take the derivative with respect to y we've been here before, after derivatives with respect to x we've received.

Here we've received the first derivative with respect to x After we've received the derivative with respect to y.

May come to mind right now, I wonder if these if there is a relationship between, there is.

This is shown in the following theorem.

Short an equivalent here alone before Let the representation.

f under the indicator, sub-indicator as x

When we put that derivatives with respect to x knew.

This means that one more derivative with respect to x f (x)

derivatives with respect to x is a more in it FXX as we show.

Now we do not use it in two xx.

again by the first derivative of y to y based on

If we take the derivative of the fyy call it, this consistent representation.

However, the partial derivative with respect to x to y If we take the partial derivatives

before x to y then it is We specify the or

During reverse it before then, according to y in that respect to x

partial derivatives in the short writing fyx is going on.

In some cases it is useful to show short, saves time.

Not in the sense of a force.

In some cases, the open letter means being easier to provide.

But both of these are the representation used.

Now here's the important terms in this complex, mixed derivatives of second order.

by x and y with respect to x here before then y

here, according to y wherein x by taken after derivatives.

Both are equivalent.

Our little one requirement, we have criterion.

We say that the function f, the first partial derivatives

This second mixture is continuously derivatives are equal.

So let's results in which we get We say do not change.

This is an important thing.

Of course, what happens if sustained, it may be unable to provide.

May equally well in spite of everything, but ensuring there is no guarantee.

He also carefully examined cases.

But almost all of us have examined This will be the continuity of functions.

And again, this continuity in practice many rate is very valid.

This is important with discontinuous.

For example, shock waves, such as supersonic a plane

some way to go supersonic speed shock waves are formed.

Discontinuities may occur there and stuff, but these more advanced topics.

When you thoroughly understand it further will not feel difficulty in understanding the subject.

This Clairout higher-order theorem Derivatization

, higher order derivatives in the There are generalizations.

For example, two derivatives with respect to x, y, according to Let's say you take a time derivative.

You can do this in three different ways.

Twice before taking derivatives with respect to x to y As you can get by once

xa, then after taking the derivative with respect to y You can get a more derivatives with respect to x.

Or after receiving the first derivative with respect to y

derivatives with respect to x twice consecutively you can get.

All of them together at all

are equal, however, that this continuity provided case.

Of course, the second partial derivatives must be continuous.

And that, regardless of the order of two times x

by once taking the derivative with respect to y We can write.

Clairout all of them equally theorem says.

Similarly, but a twice to y time derivatives with respect to x we get

If we consider the opposite case, wherein, There are three different possibilities here.

These are equal to each other again.

Twentieth order, albeit still If continuity are met

If you get taken in what order partial derivatives unchanged.

Let's do an example.

Given the function x squared minus two x plus three

E-sine squared x times y plus years to get over.

The first and second order partial We want to calculate the derivative.

Of course the second order partial derivatives perhaps may be the first step to calculate

If so desired, but the first is a natural way calculating order derivatives.

This is the second of the first order derivatives proceed to order derivatives.

Here again the first order derivatives As you know, xa

y based on partial derivatives treated as constants Retrieving.

this time in partial derivatives to y x is taken as constant.

It turns out that our results are in the example.

When we move to the second order derivatives f x of x by this again

We can take the derivative of x to y or f we can get the derivative.

Similarly, according to the f y y times

As we further variant, according to x We can take the derivative.

These accounts are simple calculations.

For example for one more time of x to x

When we take the derivative of x by two two is coming.

This constant derivative is zero.

The two sine cosine x the derivative of x As the times have also

multiply the second derivative of the first, then cosine squared is happening.

Plus the second derivative of the first times.

The second derivative, the derivative cosine minus x sinus

sine-squared is negative for coming.

the partial derivative with respect to x to y via we have received via e y remains.

However, all of them, all accounts we see.

Again Clairout theorem, i.e. If we let the order in which mixed

derivatives coming equal naturally We see that arise spontaneously.

See it here for derivative of x to y 're getting.

This time from a different function here xa

We take the derivative with respect and equal involved.

Already a need to go.

Because these functions are continuous.

Clairout theorem is continuous for available.

Clairout their equality in Theorem is already guaranteed.

We do not provide here would have.

Find the numerical values of these derivatives if we wanted, for example, x is equal to

pi divided by four, i.e., in a forty-five degree angle x for

If we choose zero for y here you can see

The forty-five degrees from the sine and cosine sinus and

equal to the cosine, square root of two divided divided by the square root of two or two

See if you can.

Here we see that almost one-half of this square root

two, one divided by the square root of two one-half gives.

This is simplified by two.

y e to be zero because zero a, an income here.

As you can see, this one by two negative combined minus one remains.

Here are two x two x two times pi From the pi divided by two divided by four remain.

Others are calculated in a similar way.

Maybe I'll give some time You can watch, but

make them do it yourself need.

Tennis or basketball sitting on a sofa or football can not be learned.

It is a good match to watch, but if you yourself

If you want to learn something in the field You need to leave.

Here is a paper and pencil to take it that the scope of accounts to make yourself.

I want to give an assignment.

This assignment, similar to a previous problem.

The same function, but this time the third-order In case you have found derivatives.

Here you will learn the second order derivatives

already found on the previous page with a.

When we received the third order derivatives As you can see with respect to x

may derivative three times, three times to y derivative.

But two derivatives with respect to x, y, according to times can be derived.

But they also come in different row You will ensure that all the same.

The complement thereof, that status symmetrical time two to y

time derivative of a time derivative, but with respect to x happens.

There are three different possibilities in this.

As well as being different.

But three of them still function continuously will have the same value for that.

You will be able to see that there's a.

They are all fully third-order partial derivatives.

So far, a variable reminders as we have seen.

We generalize to the two variables.

Now the same variable genellesek approach

ie the variables minus one We will freeze temporarily.

Hence a variable function is reduced.

And he will take the derivative of a variable also.

By definition, a function so If x a, x two,

x j x N-linked, derivatives thereof and x j based on

türevini x j'ye bir delta x j değişimi We give.

Bunu fonksiyonun değerinden çıkarıyoruz.

Demek ki bu z desek bu f değerine, düşeyde tabii bunu

hemen çizemeyiz çünkü burada üç boyuttan fazla bir boyut gerektirir, uzay lazım.

Ama biz de üç boyutlu uzayda yazıyoruz, yaşıyoruz.

Ve dolayısıyla bu fonksiyonu çizemeyiz ama pekala hesapları yapabiliriz.

Zaten bilimdeki temel ilerleme de dokunabildiğimiz, görebildiğimizin

ötesini matematik sayesinde ulaşım sağlamakla oluyor.

Özellikle n eşittir üç için, yani üç değişkenli fonksiyonda

x'in, y'nin ve z'nin fonksiyonunda x'e göre, y'ye göre ve z'ye

göre kısmi türevler aynen daha önce bildiğimiz gibi hesaplanıyor.

x'e göre kısmi türevde sadece x'e değişim veriyoruz,

bakınız buradaki y ve z'lerde bir değişim does not exist.

Ve bunu x'e göre uzaklığına bölüyoruz, yataydaki uzaklığa.

Tabii buradaki yatay çok simgesel anlamda çünkü bu dört boyutlu bir uzay

gerektiriyor x y z ve f'den hesap edilen w diyelim dördüncü boyut olmak üzere.

y’ye göre kısmi türevde ise sadece y’ye değişim veriyoruz.

z’ye göre kısmi türevde de z’ye değişim We give.

Örnek fevkalade basit.

Böyle bir fonksiyon verildiğini düşünelim.

Bunun x’e göre kısmi türevinde x’e

değişken y ve z’ler sabitmişçesine hesap we see.

Dolayısıyla x karenin türevi iki x, y karenin türevi sıfır z kübün türevi sıfır.

Neye göre?

x’e göre sıfır.

Burada y z kare var bunlar bir sabit gibi duruyor.

Bunun da x’e göre türevinden x’in x’e göre

türevi de bir olduğu için bunu elde We are.

y’ye göre türevde de tamamen benzer düşünceyle Bu sefer x kare sabit

eksi iki y, veriyor eksi y karenin türevi.

z yine sabit y’ye göre kısmi türevde, ordan bir katkı yok.

Buradaki x ve z kare, sabit görevi yapıyor bunları yazıyoruz.

y’nin de türevi bir onun için, buraya da bir geliyor.

z’ye göre kısmi türevde yine aynı düşünce x

y contribution from them square and fixed frames does not exist.

z z square cube comes three derivatives.

Here too hard, but our frame of x and y derivative

here comes two two z as z We are adding.

As you can see, this fantastically simple partial take derivatives.