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.