Chain Rule Problem Solving II DIfferentiation II Lecture 03

Added: 2026-05-23

[00:00:00]From 10 to 10, you ca n't say three numbers first, look at it, three numbers, three numbers are differentiation of root x plus one by root x, yes, then you do n't need to do anything with this root x because root x has a direct formula, plus it can be written as x to the power, how much can it be written? If you write it above, it can be written as x to the power minus half. If you differentiate root x, you will have to differentiate by two root x and by two root x and x to the power minus half. If you differentiate minus half x to the power minus half, minus one and by two root x, you will have minus half x to the power minus the by two.

[00:01:07]Now, how much is this? I said seven numbers, seven numbers are five to the power x plus two log x. This is the point. In the case of differentiation, if there are multiple quantities to be added or subtracted, we differentiate each one separately.

[00:01:32]The problem is the log, that's the problem. You can differentiate this.

[00:01:38]No base is given for the log. This has become a problem. In calculus, especially in differentiation, if no base of the log is given, the base of the log must be assumed. I mean, it's easy to take it for granted.

[00:01:51]And if the base is given, then that is the base. So it's fixed. But if no base is given, normally we are taught that if no base is given, then we assume that it is base.

[00:02:03]But in the case of calculus, if no base is given for a log, we assume that the base of the log is the base of the log. If this is the case, then it will be a single number. If we differentiate the power of x, then we will get the power of x plus two.

[00:02:20]Think of it as a number. If we differentiate the power of x, then we will get the number x.

[00:02:22]Next problem is 13. There is 13. What is the problem in 13?

[00:02:36]Differentiation of x byte two plus sec inverse x. Where is the problem here? If we do x byte two, then we will get the number byte two. Let's break it down. Differentiation of x or but plus differentiation of sec inverse x. Here, how much is the constant?

[00:03:05]What will be the half constant? What will be the half that will go ahead?

[00:03:08]Who will differentiate x? But if you do the last ex but, it will be byte plus that inverse x then direct formula one divided by x root over x square minus one next problem 18 number 18 says differentiation of cube root x differentiation of cube root x what does it mean what power of x is byte power formula byte will come in front x to the power one byte will be subtracted one byte will come in front x to the power if you continue how much will be minusboth after 18 after 18 up to 20 there are no more problems up to 30 well it was left here [snoring] how many numbers problem 41 to 50 problem tell me.

[00:04:25]41 to 50 41 to 50 45 45 is the number like this differentiation of root over x squared plus one, who would say whose formula would be first, root x, root x x, who do you think of, think of it like this, root x, if you differentiate root x, you will get one byte root x, there is no x in place of x, you have to differentiate it again, who will need to differentiate x squared plus one again, and the byte of x squared plus one will remain, if you differentiate x squared, you will get two x plus one, if you differentiate zero, eight, and eight will be cut off, there is another problem in 50, there is a number 51, the problem in 50 is number 51, differentiation of 2x x plus 5 to the power seven, who would say whose formula?

[00:06:01]Source of power. So what would be the formula for power? The power will come forward.

[00:06:06]Power will come forward. The power will be reduced by one.

[00:06:14]What is the meaning of the power formula?

[00:06:18]X to the power n you said x to the power n but it is not x then it needs to be differentiated again so if there are multiple additions or subtractions then first take a bracket and differentiate it then differentiate x then if you do that then plus j [sniff] Alhamdulillah [sniff] then say after 51 out of 60 there is 59 [sniff] the number 59 says differentiate a x squared plus b x plus c to the power n then the formula is x to the power n n will come up a x squared plus b x plus c from here n minus one will be x instead of x is there no x in this whole then this whole needs to be differentiated again do it one by one first what do you do first what do you do first who do you differentiate x squared who do you give plus what do you do bi who do you differentiate x Who did one and who did C [sniffing] Then say no more in 60 60 to 61 to 70 in 70 [sniffing] How many say 61 to 70 69 Let's do the differentiation of this is not a problem Someone said that there is a problem between two people This is whose formula is X to the power N How is X itself a power or not If X itself is a power The formula of A to the power of X This is how is X How do you think A to the power of X The formula of A to the power of X is A to the power of X The formula of Lon A gave X Who was in the place of Lon then differentiate him again Then say no more in 70 How many in 80 75 numbers An important math 75 numbers will be seen Look carefully Differentiation of Lon Listen These maths of log These maths of Lon are of this type Maths of X After addition or subtraction If there is anything to differentiate this kind of math Then, at the end, you have to make lasagna. You have to make lasagna. If you do, you will see a lot of money being spent. If you do this type of math, you will lose a number. Let's do the difference first. [Snorting] If you say " Lon X" or "Lon X", I wrote "One By By X". The one in place of X will need to be differentiated again.

[00:10:15]If you have trouble understanding, break it down.

[00:10:18][Snorting] If there is more than one addition or subtraction, each one has to be differentiated separately. For example, if we differentiate x, we get one. And whose source do you give this here? If you differentiate root x, say one by two root x, then differentiate who was in place of x.

[00:10:42]Differentiate x squared again.

[00:10:46]Differentiate x and one. If you do this, what is the difference here? What is the difference there?

[00:11:06]Root over x squared plus one. If it is a difference, then it will not be here. Root over x squared plus one. X squared plus one. There is an x ​​x plus root over x squared plus one. Root over x squared plus one. If you divide this by one, you get 75. Then tell me which one is in 80.

[00:11:29][Snoring] If there is confusion, solve that too. I think I am doing the answer to this. If it is like that, solve that too.

[00:11:39]Which one of 80 is 81 to 90? The problem is 85 numbers.

[00:11:48]85 numbers.

[00:11:53]85 numbers are like this. Differentiation of one divided by x squared plus 5 on top of it is squared.

[00:12:07]This is 85.

[00:12:10]Don't differentiate it first, bring it a little bit properly.

[00:12:16]How much minus can be written on top of x squared plus five?

[00:12:20]Then whose formula is the formula of power?

[00:12:23]What is the formula of power? The power will come up.

[00:12:28]Now, it is said that x to the power n is minus one from what is there. Now, it is said that x to the power n is not x in place of x. It will have to be differentiated again.

[00:12:37]If you do x squared and if you do five, it will be zero. The result will be minus f x and divided by x squared plus five. You can write the cube on top of it like this.

[00:12:50]Minusf x by x squared plus five.

[00:12:57]Then tell me how much 85 80 went. Which one is out of 90? Tell me which one is out of 90. Hey, you dare to say it. No problem. Between 91 and 99, 96 is the number [snorting] Before I do 96, I will give you a Let's show differentiation. Look at this first.

[00:13:25]Differentiation of e to the power minus x.

[00:13:32]Differentiating e to the power minus x gives minus e to the power minus x. I'll memorize this.

[00:13:40]Differentiating e to the power minus x gives minus e to the power minus x. Now listen to how it works. First, you would tell me whose formula is it?

[00:13:50]What is this formula? If you differentiate to the power of x, then to the power of x means what is there. So who is there in the place of x?

[00:13:59]Then you need to differentiate minus x again.

[00:14:01]Differentiation of minus x minus x what happens when you do minus x? If you do minus x, it will be minus o. So what happens when you differentiate minus e to the power of e to the power minus x, then you differentiate minus e to the power minus x. The number 96 says that you differentiate it like this: X plus e to the power minus x differentiate it and say Fast Car Formula Law's formula is o or x or x Now tell me what was in place of x. Do these two need to be done again? If we differentiate e to the power x, if we differentiate RE to the power minus x, minus e to the power minus x, then my answer will be above e to the power x minus e to the power minus x below will be e to the power x plus e to the power minus x [sniffing] Which other problem can we get 100 by removing 100?

[00:15:08]There is a multiplication formula in 100 97 [sniffing] The number 97 says the number 97 says the number 97 Differentiate the root over x plus the root over x.

[00:15:37]Fast Car Formula Root X Think about who you are thinking of first.

[00:15:42]Root X, but you think that if you differentiate this root X, then it will be o byte root X now in place of x. I will differentiate it again. In place of X, this is [sniffing] One by two root X plus root X. If we differentiate X, we will get one root X. If we differentiate X, we will get by root X. Which one is one to 99? There is no problem. Give him some more practice.

[00:16:30]After practicing, today we will look at the multiplication formula. [Snorting] I think there is something like that.

[00:16:40]Differentiation of the tangent sine e to the power x cubed. [Snorting] Differentiate it. So you have to do it one by one. We will take them one by one and differentiate them one by one.

[00:16:58]First of all, tell me whose source are you looking at?

[00:17:01]This is the formula for Holtan X.

[00:17:04]If you differentiate the extan x, it becomes square x. So now it's not like that? So, if you give the first clue, that's it. Whose source are you looking at now?

[00:17:16]Psi X. But do you remember this? Psi X. If you do X, you get X. So it's gone. Who are you looking at now? If you do To The Power X, then To The Power X will be gone. Now look who's up there? If you take x cubed, differentiate it like this, and think about it like this. There is such a thing. So whose formula did you look at first?

[00:17:54]First, if you differentiate the sine inverse x, the inverse x will be like this: by root over 1 minus x cubed squared on top of it. You didn't give the formula for this. The next one is whose formula it is that x itself is the power a to the power x. The formula for x is a to the power x.

[00:18:24]Then you gave the formula for this. Now whose formula is x cubed, it will be three.

[00:18:40]There is differentiation of root over and minus x squared. So whose formula is root x?

[00:18:47]What is the formula for root x? Byter root x means 1- x squared. Again, who needs to differentiate it? Do it one by one, if you do zero x squared, will it intersect these two and these two? Then it will be minus x. By root over and minus x squared.

[00:19:21]You are telling you that if you differentiate one plus sine to x, square it on top of it. The power formula will come in front of two. The power will come in front of plus sine to x. One will be reduced from the power of x. If these were in place of it, then it will need to be differentiated again.

[00:19:53]One plus sign to x will need to be differentiated again.

[00:19:58]Then two it two one plus sign to x will remain. Differentiate the ball. If one is done, then the site of x will be done. If the site of x is done, then the cost x will be the formula of psi x. Here, the formula of psi x will be directly written. After that, what will happen if the site of x is written? I am writing cost x. I will need to do it again. Not x will come in front. If cost x is this, then the differentiation of e to the power five is like this.

[00:21:04]If the power five is like this, then how do you do it?

[00:21:11]First, the formula of to the power of x is the formula of to the power of x. What is there?

[00:21:14]Lon k x is e to x. Who is in place of this?

[00:21:21]Then what will you do with five? It will come in front. Differentiate who is lon x? If you do by a x again, then the answer will be 5 e to the power 5. If you take a x minus the front, it will become psi x by cos x. Okay, let me tell you a little bit about how to do this math in a book.

[00:21:56]This is cancer. What you are doing is wrong. This is how the math is done in the book. E is the power of five if x is the power of e and x are together. If E and Lon are together, they rise. Do you understand? If E and Lon are together, they rise.

[00:22:18]Now to get that, there is a five five in between E and Lon. First, we have to take out the five five. From here, in the case of log, if the power is in front, it does not come forward. If it is in front, I cannot give it to the power. In the book, it is written like this: first, E to the power Lon, cos x to the power five.

[00:22:42]See if the power Lon comes together. If it comes together, it will go up. You will only have cos x to the power fi. Now differentiate it. You would say, whose formula is the formula of power. The formula of power. What fi will come forward.

[00:23:03]Who was in place of cos to the power f?

[00:23:06]If we differentiate x, both will be minus psi. You can answer in two ways. And if you want, you can take this from here.

[00:23:22]Read the differentiation of multiplication formula. [Snoring] Write the formula for the quality of differentiation [Snoring] The formula for the quality of differentiation. The multiplication formula is called UV's formula. The multiplication formula is called UV's formula.

[00:24:04]Understand it better first. Look here.

[00:24:08]Suppose you have the differentiation of e to the power x sine x.

[00:24:16]If I give you something like this, differentiate it like this, which many people do. If you differentiate E to the power X, if you differentiate X to the power X, then it will not be X.

[00:24:29]Here, what are the two quantities in this state?

[00:24:32]It is in the multiplication state. It cannot be differentiated like that. If it is in the multiplication state, there is a separate formula for multiplication, which is called the UV rule. Look at that rule.

[00:24:41]What is the rule? Differentiation of U It V U is a function V is a function. Just to explain, think of it as U This is V Two quantities are multiplied. The UV formula is like this. If two quantities are multiplied, keep the first one and differentiate it. Keep the second one and differentiate it. Keep the second one and differentiate it. This is called the UV rule. Look at it first.

[00:25:13]If you have to differentiate it, the multiplication formula is not read. I will put the two power X. I will differentiate the sign of X plus the sign of X. I will differentiate the two power X. Is there any differentiation sign in front of this. It will be like that Differentiate si x plus si x will be itudi power s [sniffing] Next look at the example look at the differentiation of x tan x x see if two quantities are in multiplication then UV formula UV formula is I will put u differentiate vtaket x plus tan x differentiate x will be x if tan x is differentiated sec square x plus tan x will be x if x is differentiated think like this differentiation of sine x e to cos x is like this then this is u this is v then UV formula is like this I will put sine x will put sine x will differentiate cos x plus cos x will differentiate sine x then sine x will remain a what will happen if x is differentiated minus sine x plus a x if si x is differentiated a x then write that first write the plus first Cos square x minus sine square x cos square x minus psi square x cost x formula I wouldn't do it like this UV formula later I wouldn't do it with UV formula I would do it like this look differentiation of was telling me if you put a two in front of sine x itu cos x trigonometry formula doesn't fall into sine x cos x half into one half nib differentiation will be done here one two nib sine x cos x half will be there I'm not doing differentiation to psi x cos x who formula site this formula now I'll differentiate if it's the multiplication formula and it's going to take a while to give site x if you differentiate cost x again who will do it x will be cut off when can it be done whatever you like whatever we were doing math on multiplication formula so look it will be a little like this [sniffing] differentiation of this todi power x sine inverse x like this If you do, then see if the two quantities are in the multiplication state. Now you can remember in your mind.

[00:28:46]Brother, yesterday too, the quantities were in the multiplication state. They were not in the multiplication state.

[00:28:50]Yesterday's quantities were like this.

[00:28:52]Differentiate them. To the power sign. Not x cube. Tell me where there are multiplications. How many quantities are there?

[00:29:00]Here, there is only one quantity. To the power x is only one quantity. [Sniffing] That means there are many quantities to differentiate, but only one has everything. [Sniffing] So, who will have to differentiate?

[00:29:19]Differentiate the sine inverse of x.

[00:29:22]There will be a plus sign inverse x. Who should be differentiated? Differentiate the curve to the power x. There will be a power X today.

[00:29:30]Differentiating psi inverse x will give us one by root over and minus x squared plus psi inverse x.

[00:29:39]Who should be differentiated? If you differentiate the Tudi Power X, you will succeed.

[00:29:42]Now tell me that I wanted to write the formula without writing the middle line, that line needs to be written. Look, what is the formula for UV? I will put U in it, I will differentiate V. I will put V in it, I will differentiate U. How do you think it is? It tells me that differentiation of X is to the power of X. So I understand that the first one is U. The next one is V. So if that is the case, what should I do? I will put X first. Who will I differentiate? If I differentiate X to the power of X, plus I will put X to the power of X, who will I differentiate?

[00:30:18]If I differentiate X, I will do it directly at once. There is no need to write the formula. Suppose it is like this: differentiation of X cube cossack. It is like this: so who will I put first?

[00:30:34]X cube cossack.

[00:30:37]If I differentiate X minus cossack cossack plus cossack. Who will I put next? Cossack x. Who will I differentiate X? If you do a cube, think about it like this: Differentiation of x is [sniffing] x is x is x. What should I do? It goes ahead. It goes to the front If we differentiate the power of X, the multiplication formula is not applied here again, then the multiplication formula is given again. X will remain. Who will we differentiate? If we differentiate the power of X, we will have plus. Who will we differentiate?

[00:32:45]If we differentiate X, the common between these two will be to the power of X. If we differentiate it to the power of X, it will be x cos x plus x sine x plus y sai [sniffing] If there are more than one quantity, we will take more than one quantity together. We will take four quantities. We will take two and two together. We will take two and two together. We will take V. Okay [sniffing] Now let's see how many quantities are there in the multiplication condition.

[00:33:31]So whose formula is UV? Then the formula of UV is like this.

[00:33:36]X square will remain. We will differentiate E to the power of cos x plus E to the power of cos x. We will differentiate Sine X squared. So tell us, Sine X squared remains. If we take X to the power of X, what is in place of X?

[00:34:03]Then we need to do it again minus psi plus to the power cos x.

[00:34:11]Whose formula is the power formula? Or the power formula would only be when there was a square here? psi [ sniffing] X is the formula psi x squared, who would we need to do it again? If x squared, who would we need to do it again?

[00:34:29]To x [sniffing] Let's say there is a formula like this. First tell us whether it is the multiplication formula or not. UV. The first is U. The next is V. Okay, so what is the multiplication formula?

[00:34:56]I will keep the first one. Who will I keep?

[00:34:59]Th to the power. Ten. Who will I differentiate?

[00:35:04]What will happen if I do lon sine x?

[00:35:10]And we need to do lon sine x? One by psi x. Again, what will happen if lon sine x? What will happen if lon sine x?

[00:35:17]Now we will keep lon sine x? Who will we differentiate?

[00:35:24]So, whose formula is this?

[00:35:28]What is the formula for lon sine x?

[00:35:34]Who was in place of A to the Power X Lon X?

[00:35:36]Do it again.

[00:35:39]That square X is like this. Whose source?

[00:35:55]Multiplication formula Multiplication formula will be the first one. To the power of x cube will be the one who will differentiate. Even if you do x square or x square again, who will need to do x square? If you do x square, you will have to x plus x square. Who will differentiate?

[00:36:14]Whose formula is to the power of x x cube again. If you do the multiplication, what will happen is that x and x square are subtracted here.

[00:36:24]This is how it goes. Okay, take the homework.

[00:36:29][Sniffing] Homework is from sheet 21, how much did it increase from 26 to 40? This is from the sheet. It is not from the book. It is not from the book. It is from the book. It is from the book. It is from the book. It is from the book. It is from the book. It is from the book. It is from the book.

[00:36:57]You were learning the chain rule.

[00:37:02]If that homework that was given is 100.

[00:37:06]If you have done some math, you will be able to say it automatically orally. There is no need to do it in the differentiation book. You can say it orally. I want to see if I can do it orally. I will do some math first. Then I will catch you. There is such a thing as differentiation of sine e to the power x cubed.

[00:37:33]How will this be differentiated? First, the formula for sine x will be. First up will be the Kai to the Power X Cube.

[00:37:42]Then the sign is finished. I'll come to this now. To the power x cubed is to the power x cubed. If you do x cube again, then the answer will be x to the power x cube. If you do x cube again, which one is after which one?

[00:38:06]If you have trouble grasping this, what you can do is you can cut it in your mind. How do you hold it like this?

[00:38:20]Hold the sign like this. It is to the power 5v x like this.

[00:38:24]What you can do is you can cut it in your mind. How do you cut it? First, whose formula is it? First, hold the extan. This whole extan. If you do x, it will be square x.

[00:38:43]So, let's cut it. Think about it, the formula for extan is over. Now, whose formula is it? If you take 5x to the power, what will happen to the power? If you take 5x to the power, what will happen to the power?

[00:39:01]Now, if you take 5x to the power, do it step by step one by one.

[00:39:10]Stop, stop, stop, stop, stop the sound.

[00:39:26]Now, take that speed up. Now, let's talk a little bit.

[00:40:14]First, what will happen to the formula? If you take 5x to the power, what will happen to the power Minus sign tan to the power x se square e to the power x again to the power x first the formula of root x but root sai lon then sai formula it byte root to the power five x e to the power five x five x k The formula for to the power of x is to the power of sine to the power of lon cut to the power of lon cos x sec square e to the power of lon a x e to the power of lon by a my sai x ok okay tell me by lon it to lon x k again or x it is also by sine x square a x squared x this is whose formula a the formula for to the power of x is a to the power of lon to the power of x who gave the formula for it this time kelon sai x k or sai x lan formula for lon now if sai x k k this is the end of this is whose formula a to the power of x is a to the power of x who gave the formula for lon a sai x k if sai x k k this is the end of this is whose formula a to the power of x is a to the power of x who gave the formula for lon a sai x k if sai x k by where sai x minus is sent to the front these two together would write sai x or cos x say whose formula is sai inverse x then the formula is o minus x squared over the square at the end of this again No need to square x, what will happen or will it be negative sign then it will be negative tan x, I did this the other day, I think it's lon x by x, if you differentiate e to the power x, you get e to the power x, if you differentiate e to the power minus x, you get minus e to the power minus x, who told you the formula?

[00:47:15]Sai is not inverse. But the source of power. The power will come in front of the psi inverse x. This will need to be done again by root over minus x squared.

[00:47:37]Whose formula?

[00:47:40]If you differentiate the inverse of x, but if there is x in place of x, then it will go up to the power x one plus it can't be written like this.

[00:48:07]Whose formula would you say? I will not give the formula for the sine inverse of x.

[00:48:15]This is the inverse trigonometric function. This is a trigonometric function. It doesn't cut. I mean, what else can you get up to? It can never be cut like that.

[00:48:24]So the differentiation of these two will be eliminated immediately. It will only be X when you get up. Even if we differentiate X.

[00:48:34]Suppose I said these two are cut off.

[00:48:37]After leaving, Rifat released a video like the Academy.

[00:48:39]Those two never intersect.

[00:48:43]If psi is like this, sine inverse sine x, it also means x.

[00:48:54]Sine sine inverse x also means x.

[00:48:58]A trigonometric function. An inverse trigonometric function.

[00:49:05]Tell him.

[00:49:07]Whose source?

[00:49:09]Look, stop writing for a moment. Stop talking.

[00:49:13]This is A to the power of X. The formula for A to the power of X is A to the power of A to the power of X.

[00:49:31]You gave the formula for this.

[00:49:34]Now if we do lon tan x, we will get the formula for bytan x.

[00:49:44]Differentiating tan x, we will get sec square x.

[00:49:48]Now listen to the talk. Observe this term a little bit. Instead of bytan x into sec square x and bytan x into sec square x, look a little bit and by tan x into sec square x.

[00:50:13]If I break it down, its answer is cos cot x. Look at it. Take cut, cut, then we can write cot x as x divided by sine x sec square x. We can write cos square x as cos x cos square x.

[00:50:38]See if there are sine x and cos x in front of it. Do we need two? I am taking one two above, and one two below, and taking one two in front of them.

[00:50:51]sine x cos x two left 2. sine x cos x means sine x two left 2. What does 1 / sin mean? The cosac 1 / sin means cosac. So 1 / sin 2x means cos 2x. So this term is actually 1 / tan x * cos square x. In fact, we will directly substitute 2 cos x from now on. I'll explain it to you then. 2 What was the angle here, Koskot X?

[00:51:25]And here the angle has doubled to x means if you ever come to this by ten hold like this comes five x into it is multiplied with sec square of x then its angle will be cosec x or hold like this comes to you and by tenth x into sec square 3 x in that place we will put it directly cosec we will multiply this angle by two just how much x 6 x lift it up a little ok then the angle of this will be ton to the power lon ten x lonton it will be written to cosec into which will be doubled x tot in front okay okay tell me whose formula who did minus sign to the power ten a square to x i gave the formula of e e to the power x if we differentiate it will be to the power x i gave the formula of power what not x i gave the formula of power now tan formula tan x i gave the formula of square tan who did this four in front I'll take it. Next, you can't always do the differentiation that you see directly. This work cannot be done. In some cases, we will minimize it first.

[00:54:39]After minimizing, I will differentiate. It will be done directly.

[00:54:43]I'm not saying that something like that won't happen. We'll shorten it first. The formula for differentiation of trigonometry and the formula for minus cos x is sine square x minus one plus cos x. The formula for cos square x minus two is sine square x minus two. If you divide two by two, will you divide sine square by cos square by square? Will the square x minus square and root be equal?

[00:55:19]So what will happen?

[00:55:21]Differentiation of Longan x Byte. It will happen. Now I'll tell you to differentiate. The formula for lon is lontan x byte2 and by tan x byte2. If you do lontan x byte2, you will get sec square x byte2. If you do lontan x byte2 again, you will get byte2. Does this look familiar in any way?

[00:55:56]Its angle is squared. Will this angle be doubled if you multiply it by two?

[00:56:03]What happens if you multiply it by two? X is squared. There are two and two are cosecants. Take this. Okay.

[00:56:22]Next, you understand that trigonometry is very important.

[00:56:32]I am giving you a division by b. I am giving you a division by b. Is that a question that you have seen in life?

[00:56:44]How many divisions are there in the question? There are four categories.

[00:56:49]Section A, Section B. Okay?

[00:56:53]How many questions are there in section B than in section A? Four o'clock. There are four questions in section B. How many questions? Eight o'clock.

[00:56:59]How many answers do you have to give? You have to answer five. I am teaching you the entire B section. Section B includes trigonometry and calculus.

[00:57:12]Trigonometry and Calculus There will be two sets of trigonometry exercises from calculus. There will be two sets of trigonometry exercises from calculus.

[00:57:26]How many chapters will there be in calculus exercises?

[00:57:30]How many differentiations will there be? There will be four questions from these four chapters. So, now you will not be able to answer any of these four.

[00:57:43]What if? What am I teaching you first?

[00:57:46]I was talking about trigonometry, didn't I? If you do n't know trigonometry, you wo n't be able to complete differentiation correctly. And if you ca n't differentiate, you can't integrate. And if you can't integrate, you wo n't be able to pass Section B. Is the matter clear? That means where is the source?

[00:58:07]From trigonometry. So I was saying, read.

[00:58:12]I remember this, whose source is this?

[00:58:22]2 sine theta cos theta is the sine. You have to observe this. The angle after the sine. Is there a same angle after the cos?

[00:58:32]Stop writing 2 sine theta cos theta sine to the theta.

[00:58:36]Look at the math. Tell me fast. What is the formula for power? What is the formula for power? The power will come in front. The power will be reduced by one from the power. To the sign, lon se x. I have given this formula for power. Now, if you do the formula for sine x, you can do lon sec x. I have given the formula for sine x. If you do lon sec x, you can do lon sec x. I have given the formula for lon sec x. If you differentiate x, you can do it.

[00:59:23]This and this intersect. Now, you have to observe this. See what is the angle after the sine?

[00:59:32]What is the angle after the cos? After the sine, the angle after the cos? After the same angle, there is a two.

[00:59:39]Then see if the 2 sine theta cos theti formula is followed. Is there another one at the end? You have to do it like this.

[00:59:54]Look at one more and then I will pick it up.

[00:59:57]It was said that first whose formula is first the power formula what is the power formula will come in front of you si lon x square then the power formula is given now whose formula is si x if you take lon x square sine a formula is given if you take lon x square or x to the power for and by x square now if you differentiate x square this will happen now notice that after the sine there is a two in front of the same angle then to sine theta cos theta whose formula is sine to it theta means if you cut lon x square then there is 2 by what is done in the book there is another work done in the case of log if there is power that power comes in front this power comes in front how much will it be when multiplied by two that or x in front I wrote the sign of this answer is the result lon x this has been done take it up properties of logarithms or properties of log so here you will get five properties of log in total and five properties of lon.

[01:01:53]Look, what happens if you say log x to the power n of a number?

[01:02:05]What happens if there is power in the case of logs?

[01:02:08]N log x, of course, in the case of log, if the power comes first, then what happens if it is in front goes to the power, both are possible. Suppose you have log x cube like this, then you know what happens if the power comes first, 3 log x is the two are equal, that means sometimes if it is 3 log x, in that case, but I can take the power that is in front to the power. It's not like this always happens. Even if we are ahead in some cases, we will take it to the next level. Well, if there is one power, it comes forward.

[01:02:53]What happens if you multiply two numbers?

[01:02:56]If there are multiple quantities after the log, they can be written by adding them with separate logs.

[01:03:07]What happens if we divide log A by log B?

[01:03:11]What happens if this is a division? So will it be log A minus log B?

[01:03:21]You know these three. If there is power, it comes forward.

[01:03:25]If there is multiplication, there is addition. It is possible if there is a share. Now look at the number four first and then take it.

[01:03:31]Number four log AB I need to interchange these two, meaning B will come down and A will go up. I just do one by one.

[01:03:43]If I do one by one, log beta will come down. This will come here.

[01:03:51]When is this needed? Notice that you do n't know the formula for differentiation of log A x.

[01:04:01]Differentiating log A x will be one by x. Now think about what is given to you in the book, which is differentiation of log X. It should be like this. Should n't these two just be interchanged? How do you interchange it?

[01:04:24]And by log A x? Then what we will do is differentiation of. We can write it like this: log A x to the power minus. And then we will differentiate. First, whose formula will be read first.

[01:04:39]First, the power formula will be read instead of X.

[01:04:41]Or, we will differentiate it again. We will see later.

[01:04:45]Now, come on. The most important property of log that you need to know is that look at this. Number five log AB is like this.

[01:05:00]This is in the form of the product of multiple logs. It can be written.

[01:05:06]I will just write multiples as a product of two.

[01:05:09]Look, it can be broken down and written like this.

[01:05:13]This is the log you have kept so far. I've kept the log up to this point. So far, let's take any constant here.

[01:05:20]We will take any one here. I will always take it. I kept the log until now. I'll take it here.

[01:05:28]Into log that it will come down beta will come here let's understand see you have logthof logthof we will break it down like this log th will be here take e log now it will come down e here will be fi it can be broken down like this look again let's say you have like this log x pull x think it is like this break it down you will break it down like this log x will be here take log here will be here understand now I will see more shortcuts for this thing I will not remember this for now look here log ab is ab is where are you from when are you taking classes since four days how many months have you not come before that two months have you not come just okay log ab I asked you brother this convert this to total lon. Convert from log to lon. I'll take a good look. When is the lawn?

[01:07:02]What if there is a ring on the log? There is.

[01:07:08]I kept the log until now. What should I take here?

[01:07:12]Into LogIB.

[01:07:18]Look, there's no log base here. Is it a lawn? This base shouldn't be here.

[01:07:26]If I want to interchange it, I can write it as a one-by. Oh, bye Logi and this is Logi.

[01:07:36]So what is it? No, Lonbi. Then I can write the above, Lonbi. And can I write the following? Take a look and see that log AB can be written as lonB by lon. I am writing here. It can be written as lon b by lon a, not before, not before, not before, listen first, log ab. If I say convert lon a, then this jb will be up, not up, since lon b will be down, so lon a will be down. That is, if I tell you that lon is sorry log x sine x, convert it to lon, it will be up, lon sine x, and we can convert it down directly.

[01:08:23]I'm telling you, convert it into a lawn. The top will be the lawn or the bottom will be the lawn. Take this.

[01:08:38]Five characteristics of a log. Now let's talk about the lawn.

[01:08:43]Look at some of the characteristics of a lawn. When is the lawn mowing? The log of e based on log, which is the base of log if e is what we call it, lon x, the two numbers lon and e are ever together, lon e to the power x, lon re together will come out just answer will be x, that is, if you ever see like this, lon e to the power sign x, lon r will go away, only my sign x will remain and the same thing happens in the case of e and lon, if this happens in the case of e and lon, lon x, lon and lon together will come out, whatever is in the power will remain x.

[01:09:34]What is the value of the fourth number lon? The value of lon is one and another thing to remember is that the value of lon one is z. These are some of the things you need to remember in the case of lon. Okay.

[01:10:15]Now look at the math, stop writing, listen here, first I will differentiate it, see the function, I will differentiate this quantity, whose formula is e to the power of x, which is the formula of what is there, so I differentiated e to the power of x, then why is that in front of me, what will happen in the form of power, I will give it in front of me, whose formula is lon x, and if you differentiate bytefive x, I gave the formula of lon x in front of me, if you do x, then square 5 x 5 x, if you do five, take a look, it looks familiar, five and multiply it by two, it will be 10 e to the power of lon to tan five x, its answer is cosquat x, then bring two to the front, 20 e to the power of lon tan five x, it is cosquat x, it is the answer, but in the book it is not answered like this, in the book it is done, pay attention, what is done in the book is that if any other property of log can be used, it uses that property It is shortened first.

[01:12:15]Differentiation is not done beforehand.

[01:12:18]If there is a feature of the log that can be used, I will use it. What kind?

[01:12:24]If E and Lon are together, they will be cut.

[01:12:28]Not together. Who is the troublemaker?

[01:12:33]Well, look, isn't it in front of Toulon? In the case of log, for example, if there is power, it comes in front. If there is power, then if there is power, can I take it to power? If I take it to power, will E and Lon come together?

[01:12:44]First, without differentiation, I will apply the properties of log, which properties are E, Lon, Ten5x, Since there were two in front, I am bringing it to power. E and Lon are together.

[01:13:01]If E and Lon are together, will there be Ten5x? On top of that, someone would say that the formula for power is Ten5x sec squared fivex fivex fivex then the answer would be Ten5x into Ten5x sec squared 5x. This is the answer. Again, you pay attention to this. In this, in this, in the book, if I do it, what is in between E and Lon? I do not differentiate first.

[01:13:44]I send it first. Lon is not X to the power.

[01:13:51]And Lon is together. Since E and Lon are together, it will be Ten5x to the power. Who would say that the formula for power is Ten5x to the power?

[01:14:07]Four again, who needs to do it, square x, this is the answer, do it according to the rules of the book, if there is someone in the middle, give him the power, take it, look at it, it would say, what formula is log a x log a, think of it as x log a x, if you differentiate it, it will be one by x x, what does o by x lon a mean, what is in place of x, differentiate th x again, then three and three will be cut off, then the answer is o by x lon, come here, tell me, is the problem reversed here? If there is any such problem with the log, you should directly convert it to LN first. So tell me if I can write it this way. Can I do this on the lawn? This time, the lawn is a constant.

[01:15:28]What will be the constant? The differentiation of will go ahead. What is left is one by lon x? I ca n't write lon x to the power minus one lon a. It's left is left.

[01:15:42]Whose formula is the formula for power? The power will come forward. If one is subtracted from the power, it will be minus two.

[01:15:54]You are saying x to the power n. The formula for power. Don't we need to do it again? If we do lon x or then I put the minus, it's left is left. It can be written as one by lon x on top of it, square in. Or is left is left above and left is left below is left is left x on top of it, square on top of it. Sir, there is probably a mistake in the number of digits in the book. It seems that x is not given here. It was in an edition like this. Well, anyway, there is a math in the book. Let me show you a little bit that the differentiation of log a x plus log x is like this. If there is more than one quantity to be added or subtracted, then I can differentiate each one separately, then it can be written as log a x plus If differentiation is possible, can it be written as log x?

[01:16:58]Then what formula is this?

[01:17:02]Direct source. Differentiating log x is also done by x lon a. How do you do this? Will you do it like this?

[01:17:15]What will be the sign? Minus will be above lon A and below will be x lon x squared above that. I'll do that and then set the value. I'll show you another [sniffing][clearing throat], just take a look. Pay attention to the log extan x. Is there any problem with the log? If you feel any problem, take it to the lawn.

[01:17:46]Differentiation of If I don't take it to the lawn, what will happen above? Lawn x will happen below. Lawn x.

[01:17:54]This is the formula for division. The formula for division will be taught in the next class. And the homework is nine, one, two, four, five, okay.