 So we are now going to start with differentiation slash derivatives. Now, first let me resolve this confusion that many students have. Many students have asked me in the past, sir, what is the difference between the two? And even my typist, he also get confused. He says, sir, the chapter says differentiation. But you have asked me to give questions from derivatives, right? So of course, he's a typist. He is not well aware of the difference. See differentiation and derivatives. What is the difference between these two? However, we use both of these letters or both of these words to represent the name of the chapter. Differentiation is a process. Okay, it's a process in calculus. And by the process, you generate an outcome, which is called the derivative. Okay. So derivative, of course, this word derivative itself is an incomplete word. It's a derivative with respect to something. Okay. But just talking about the word derivative means something derived from something which is an outcome. Outcome of what? A process. What is that process? Differentiation. So differentiation is a process whose outcome is a derivative. Okay. So in your school textbooks, in your J books, which you are referring to, they can use the word differentiation. They can also use the word derivatives. It is one and the same thing. There is no difference, at least with respect to the topics involved in that. Okay. Now understand it. What is the meaning of derivative of a function with respect to a variable? Let's say I have a function. Okay. So this function is made up of the independent variable X. That means this is the input to the function. Okay, let us say, and this is the output coming out by putting this input in a machine, which you are calling as a function. Okay. So when I say I want to find out the derivative of this function with respect to X. Let us first understand the literal English meaning. What is the meaning of this? It means that derivative of a function with respect to X is if you change your input, let's say by a very, very small quantity. Okay. Well, this quantity is a infinitismal quantity. So infinitismal is a word which is made up of two words, infinite, small. That is commonly called infinitismal. Okay. So because of that, there would be some change produced in the output. Okay. So derivative of a function with respect to X is basically nothing but what is the change in the output? What was the change in the input provided this change is infinitismal. So provided this change should be, it must be infinitismal. Okay. Any change ratio of change of output by changing the input will not be called as a derivative. For example, let us say you are moving from Bangalore to Mysore. The change is a change of let's say 150 kilometers and you are carrying out this change in let's say five hours. So change in out, change in your displacement by change in the time. You cannot say it is the derivative or displacement with respect to time, because here time is not an infinitismal quantity. It is five hours. It's a big unit of time. Okay. So what if I say, change in your displacement when you're allowed to move only from 4.000000001 vår mean that changing the displacement by changing the time that can be called as derivative of displacement with respect to time. So this infinitismal word is very important. Right. So when I say derivative of a function with respect to let's say a function of X. also, it could be function of time also whatever. If you say different derivative of a function in any variable with respect to that given variable in which that function has been written is nothing but it's the change in the output by change in that input provided that input is very, very small. Is it fine? Okay. So if I have to write it down as an expression which I have used over here, I would say, what is the change in the output? So from f of x, it became f of x plus delta x, isn't it? So this is the change in the output by change in the input. So earlier it was x, now it has become x plus delta x. So this is the change in the input where this change is very, very small. That means this is tending to zero. So as you can see the limit concept gets involved over here. So this is what we call as the derivative of the function written as df of x by dx. Okay. Here d stands for differential. Differential means a very small change. Different. Differential. Different means different is very, very small. Okay. So this change by this change is what we call as derivative of the function with respect to x. We use different names for it. Sometimes in the books, they will let it as f dash x also. Okay. Sometimes they will say it as dy by dx also. They all mean the same thing. Is it fine? Is it fine? Okay. But this expression that you see over here, let me write it in a more easy to understand way. First note this down. This is the meaning of derivative of a function with respect to x. So now later on, when we go to class 12, we'll also talk about derivative of a function with respect to another function or derivative of a function in x with respect to y also with respect to a different variable also. So all those complexities will slowly, slowly, slowly creep into the chapter. But not in this year maybe. Maybe in the next year we'll talk about it. Is it fine? Is the meaning of derivative of a function with respect to x clear to everybody? Then only I can proceed further. Now here, a very important thing that normally we miss telling grade 11th. Okay. Is that when you're finding the derivative of a function with respect to x? First of all, what are you trying to find from this? What does this actually give you in a geometrical sense? Okay. So let us try to understand that. In fact, many a times in the grade 11th, I have observed that the school teachers, they will not tell the geometrical meaning. They will directly tell first principles method of derivatives and they'll move on. But you must relate your understanding of derivatives in a geometrical sense. Then only you will be able to find the applications for this concept. So see, let me just now give you the geometrical meaning of this. So what are the meaning of dy by dx geometrically? Geometrical interpretation of derivative of a function, derivative of a function with respect to x. Okay. So let us say there is a function f of x and I'll just make a graph out of it. Let's say the graph is like this. Okay, I'm just making some orbit graphs. Okay. So let's say at a value x, okay, the output of this function was f of x. Okay. So this is your y axis graph. This is your x axis graph. Let's say you decided to change the input by a delta x amount. Okay. And because of this, because of this, your output gets changed to f of x plus delta x. Okay. So what has happened here? Earlier your input was x, so output was f of x. Now your input has become x plus delta x, so your output has become, right? So now what are you doing? By finding the derivative, you are seeing the ratio of change in the output by the change in the input when this change is very, very small, isn't it? So what are you doing here? Let me just do some constructions over here. Let us say I connect this point to this point. Let me call this point as a, this point as a b. Okay. So you can see I have created a triangle over here, a, b, c, where b, c is nothing but it is the change in the output, isn't it? So this is the change in the output and a, c is the change in the input, change in the input. Okay. And I'm assuming this change is very, very small. Then your dy by dx or your derivative of the function at x is nothing but it's nothing but it's bc by ac provided your ac length is tending to zero. In other words, it is nothing but it is the slope of a, b line. Okay. It is the slope of a, b provided your ac is tending to zero. Now please understand when your ac is very, very small, this point b is almost in the neighborhood of a, means very, very close to a. In the diagram, it seems to be far apart. It is just for you to see it. If I make b very close to a, you will not be able to see it also. Okay. So in reality, this b point is very close to a and it is so close that this slope of a, b will become actually the slope of tangent at x to the function. That means if I draw a tangent here, let me draw it like this. Okay. This tangent slope, this tangent slope, so slope of this tangent will be nothing but the derivative of the function f of x with respect to x. Is it clear? Now, here there is one very, very big assumption which I have made, which even the school teachers don't tell. The assumption here is that there is a unique tangent at x. Okay. So let me add this word over here so that you are aware of what is my assumption. So I have basically found out the slope of the unique tangent. Now, what are the meaning of the word unique? Unique means the only tangent possible at that point a. Right? At that point x comma f of x, isn't it? That is your point a, isn't it? But what are the, what if the curve did not have a unique tangent at that point? You must be wondering, sir, when can that happen when I don't have a unique tangent at a point? Okay. Let me show you a case. So let me show you a case. What if the curve was having this kind of a diagram? Okay. So let's say this is your x point. Okay. And this is your f of x. Okay. Now, what will happen at this point? There could be two tangents possible. One can be like this. One could be like this. Okay. So in this case, you see that there are two tangents possible. So there is no unique tangent. There are two tangents possible. So there is no unique tangent. Right? Then what will happen to this definition? Will this definition still be valid? Will this definition still be valid? The answer to that is no. In this case, in this case, the definition will fail. And for such cases, we will have to say the derivative of the function at such a point A does not exist. Okay. Now, this is something which in school, the teachers do not tell the students because it's not because they try to hide it from you because there is a separate chapter based on this in class 12, which is called continuity and differentiability. But remember this from class 11 only that if you're finding the derivative of a function at some point X, it is only because the function is differentiable at that point X. You cannot find derivative for any function at any Tom, Dick and Harry point and find the result. Okay. So when you're learning this chapter, when you're learning this chapter, this chapter comes with a disclaimer in class 11 that whatever function will be given to you, they will be differentiable at all points on it. But in reality, the actual picture, you will get in class 12, where you would realize that a function may not be differentiable at every point on it. That means I may not be able to sketch a unique tangent at every point of the function. And if I'm not able to sketch a unique tangent at every point of the function, I will not be able to find out the slope of that tangent or slope of that unique tangent. And hence, I will not be able to find the derivative at that point. Right. Now, as of now, a lot of questions will be arising in your mind that oh, sir, if there is a sharp point, this is called a sharp point normally, is the function not differentiable at such point, I will say yes, but that is not the complete picture. There are more things to it. We'll talk about it when we go to class 12. In fact, when you go to class 12, I have only passed my class 12 15 years back. Is it fine? Any questions? Any questions? Okay. So in class 11, the disclaimer is that you will be given functions, which will be differentiable at every point. And hence, you will always get a unique tangent wherever you are trying to find out the derivative. And the slope of that unique tangent is your value of the derivative at that point. Is that clear? Is that clear? So if let's say this point was a comma something, let's say a comma f of a, then the slope of this tangent will become derivative calculated at a, which is called f of a. So I'll come to this also. So f dash x is a basically a generic expression for the derivative at any point x. But sometimes the question center may also specify that I wanted at x equal to let's say 5. So you have to find what is the slope of the tangent drawn to that curve where your x coordinate is 5. So that will be called f dash 5. Is it fine? So this is a generic expression which I'm writing. That means x could be any point on that curve. But many a times the question center may also ask you for a specific point. Okay. So before I move on this page to the next page, is there any conceptual clarity that you want from me with respect to whatever we have discussed? If yes, please immediately highlight it. The reason me, if you don't, then you will have a tough time in calculus based topics, which is going to be 70% of your subject matter in class 12. That means 70% of the topics in class 12 is based on calculus. So you want to play safe in class 12, get your doubts clarified from now onwards. Don't wait for after class doubts. Okay. So everybody is fine. All right. Okay. So now I will be doing some discussion on this formula, which I have written over here, this formula. Okay, which I'm showing with the circle. Let's talk about that formula. So we learned that the derivative of a function can be obtained by use of this limit concept. So change in output by change in input when the change is very, very small or infinitesimal. I will just, you know, write it in a slightly aesthetic way. Instead of Delta X, I will start using H. Why? Simple. Instead of writing two symbols, I will write only one. Okay. And of course, I will simplify this and write it as H. Okay. Now, this expression, what you see is called evaluating or this process is what we call evaluating derivative by first principle. So this is evaluating your derivative by first principle or ab initio, first principle or also called as ab initio method. Ab initio or first principle means from the very basic idea of what is a derivative, you are trying to find the derivative. Okay. So please note, this is a general derivative expression for any point X on the curve. But yes, it comes with an assumption that provided provided the function is differentiable. Okay. This is an un-said thing that normally you have to keep in your mind. So if you want to find the derivative at some point A, then also this formula could be used. Okay. Instead of X, you have to put an A. Okay. So this is what we call as derivative of the function at X equal to A. Books will write it like this also sometimes. Okay. They'll put a long dash and they'll put X equal to A. So you have to read this as derivative of the function at X equal to A. Okay. Is it fine? Okay. Now something more deeper into this aspect, which again is not told to you in the school curriculum. See, this expression that you see over here, this expression that you see over here, here your H is tending to 0. Okay. So let's say if your H is made positive. Okay. Tending to 0 means it could be tending to 0 from left side also and tending to 0 from right side also. Right. But let us say if I categorically made make my H a positive small quantity. Okay. And then I use the same expression. Please note that this expression does not give you the derivative, but it gives you something called the right hand derivative. Just like you get right hand limit. This is called the right hand derivative. Also called RHD. Okay. Many books will also call it as progressive derivative. Okay. And if you write an expression like this. So I'm just showing different, you know, versions of whatever you have learned. If you write this expression like this, see, this is also a change, change in the output by changing the input. So let's say from f of x comma f of x, you're going backward. Let's say to x minus h comma f of x minus h. Let me show you in a diagram. So let's say this was your function. Right. So from x, let's say this point x and this is your f of x, you're going backward. Let's say you're going here. This is x minus h and this is your f of x minus h. Okay. And you're trying to see what is the change in the output by changing the input by using the very same formula. But remember here your H is a positive small quantity. So it is just showing you that you're going backward. Then this expression actually will give you something which we call as the left hand derivative. Okay. Also called as regressive derivative. Okay. Now, if the function is differentiable, if the function is differentiable at all points on it, then only these three terms which I have written this term, this term and this term will be equal to each other else not. That is to say, if the function is differentiable, then only your limit as h tends to zero of f of x plus h minus f of x by h will be equal to limit h tending to zero plus will be equal to this limit. Okay. So your derivative of the function will only set to exist if it matches with the left hand derivative and the right hand derivative. By the way, this also is sometimes written as f dash x plus and this sometimes is written as f dash x minus. See, left hand derivative means instead of taking a point to the right of x and then bringing that point close to x, you are doing it now in this case by taking a point to the left of x. As you can see here, I have taken, let's say this is your a and this is your b, b is to the left of a. And then I am saying how much is the change in the output by change in the input. Okay. If you are finding it in that way, you are actually finding left hand derivatives. What's the point? If you are taking a point to the right of it, let's say here, let's say I reach a point c and then you are saying what is the change in the output by the change in the input, then you are finding the right hand derivative. No, Nikhil, if your h will be zero minus, then how will you show that you are going back, my dear? Will you call it as differentiable if the state line parallel to the y-axis? No. Shall we just answer your question? Answer is no. Okay. We'll talk about all those things. A function is not differentiable at a point if it has got a vertical tangent, even though it is a unique tangent, if the tangent is vertical, vertical means parallel to y-axis, it is still not said to be differentiable at that point. But these are all things we are going to talk about in class terms. Okay. As of now, let's understand the basics here. Clear everybody? Okay. So our main agenda here or my main agenda here was to introduce you to, first of all, the most simplest expression for finding the derivative of a function by first principles. Now, when I say simplest, it can be complicated by the examiners. I'll show you in some time how it can be complicated. Okay. And apart from this, we also understood that if I'm using this kind of an expression, this kind of an expression, it gives me the right hand derivative. And if I'm using this kind of an expression, it gives me the left-hand derivative. And only when the left-hand derivative, right-hand derivative are equal, only when the left-hand derivative, right-hand derivative are equal, then only the derivative of the function will exist at that point. That means the function will be differentiable at that point. Else not. Okay. However, this part is not going to be immediately needed for you. Maybe in 12, we'll need it once again. Okay. Now, as I told you, this is the simplest expression for finding the derivative of a function by first principles. That means there could be complicated expressions also. So what kind of a complicated expression? Let me ask this as a question to you. Meanwhile, have you all copied this down? Any question anybody has? Okay. Any exam in NPSHSR? Any tests in NPSHSR tomorrow? Anybody is from NPSHSR? Shreya, you are from NPSHSR? Shreya or Noel? Because I cannot see any NPSHSR right now. Anybody from HSR over here? NPSHSR. So now, okay, you are from DPS East, Shreya. Okay. Okay. Anyways. So here is a question for all of you. This actually came in one of the exams. Not the exact question. It is just a question which I am slightly tweaked version of what actually came. Which of the following represents the derivative of f of x with respect to x? Option A. Option B. Option C. Option D. Okay. Now, this is a multiple option. Correct question. Tell me which of the options are correct. Okay. See, guys, when I say I'm finding the derivative of a function by first principles, you should only watch out for expression that says change in output by change in input when the change is very, very small. Now, let me ask you, does the first definition satisfy that definition? Does the first expression satisfy that definition? So see, this was the input first, this was the input first, and now the input was made to this. So this is the change in the output by changing the input. If you see this two edges, actually this, so is it satisfying the definition? And is this change very, very small? Yes. If h is tending to zero, so is two h. So this is an expression for finding the derivative by first principles. So even this is going to give you the answer. Don't worry. Getting the point. Tell me, does the second definition also fit the same requirement or does the second expression also fit the same definition? Change in the output by changing the input. Can I say this two h is like x plus h minus x minus h? Does it satisfy? Yes, it does. So even this is my answer. Why no? It does. Two h is what? x plus h minus x minus h, right? No issues. Yeah, what about the third one? Even third one does. So see, get this clear in your mind. Don't have a very narrow picture. Most of us, because we are solving so many school type of problems, we start getting a narrow picture of the situation. Each of them is basically the derivative by first principles. Any one of them can give you the result. What about the last one? Change in output is fine, but the change in the input is three h cube, not this. So this is not going to give you the derivative. So a, b, c are the right options here. Okay, so this understanding is very important. A question has been asked on this because the examiner knows that everybody thinks f of x plus h minus f of x by h tending to zero. That is the only expression for finding the derivative of a function by first principles. No, that is the simplest expression or that is the simplest first principle expression. You can have any one of these three doing the work for you. They will all work fine. Are you getting my point? Okay, so in today's session, we are primarily going to talk about the use of first principles in finding the derivative in finding the derivative. Okay, you may use any one of the expression that you feel like it is your call, but normally people use the simplest of all. So we'll take some expressions here. Okay, let me just pull out my differentiate each by first principle. You don't have to do everything. Maybe we'll take up the first one as of now. Okay, so please find the derivative of tan x by first principles. Okay, let's do the first one and for a change, I would use a different kind of an expression that probably many of you don't use it. I mean, many of you would not have seen the news. I will use this f of x plus h, f of x minus h by two h. Okay, the one which we discussed a little while ago. Okay, let's see whether I get my answer by using this formula. So we all know what is the derivative of tan x? What is the derivative of tan x? What is the derivative of tan x? Write it down in the chat box. Quick, quick, quick, quick, ckx. Oh, c can square x, right? Okay, so let's do this. This is your f of x, by the way. This is your f of x. So what is f of x plus h, correct? So x is replaced with x plus h, that's it. What is f of x minus h? Here, I will tell you something very important. Maybe we have done this in trigonometry. There is an easy way to simplify. I mean, there's a formula for tan a minus tan b. I don't know whether you people are aware of it. It normally comes with a little bit of practice. So if you have practiced enough question, this is the formula of tan a minus tan b. I mean, not a regular formula, which you will see in your textbooks, but this is a derived result, which normally is very helpful in solving questions. Okay, so tan a minus tan b is sin a minus b by cos a cos b. Now, what is the proof for this? Very simple. The proof is very, very simple. Just use your tan expansion. So I'll just write LHS here. So tan is sin by cos. So if you take a cross multiplication, it becomes sin a cos b minus cos a sin b upon cos a cos b, which is actually nothing but sin of a minus b by cos a cos b. See, if you remember it, you save a little bit of time. Okay, so I'll be using that. So I'll be treating this to be a, this to be b, so tan a minus tan b. So let's, let's write limit h tending to zero, sin a minus b. Now, a minus b means this, okay, into cos a cos b. So this becomes sin of 2h by 2h cos a cos b. Okay, now let us try to solve this. As h tends to zero, sin 2h by 2h. What will happen to this? Tell me, you have all done your limits very well. What is this result? One though, correct? This will be as good as cos x into cos x. So can I say this whole thing will become 1 by cos square x. See, same answer we got. We could, we would not get a different answer. You know, if you do in the regular first principle method, I think everybody has done this derivative chapter in school, right? Is there anybody for whom this chapter has not been taken up in school? Okay, most of you have already done it. Okay. So I have shown you a different way, you know, to get the derivative, not by the conventional way, but by some different definition of first principles. Okay. All right, so let's take one more. We'll take up, we'll take up the fifth one. Let's do it on the next page. I would request you to do the fifth one. Find the derivative of sin x by x by first principles. Let me know once you're done. This time, let us use the simplest expression because you don't want to complicate the situation. Okay, let's use this x derivative. f of x plus h will be sin x plus h by x plus h minus f of x by h. So this is going to become x sin x plus h minus x plus h sin x upon, so far so good. Any questions? Any concerns? Okay. Now all of you, please pay attention. I have also told this to you while we were doing the concept of limits. The factors which don't become a zero, those factors are harmless factors and you should always put the value of, you know, whatever the variable tending to and pull them out of the expression. Right now, if you see, of course, this is not a harmless factor because it will become zero when the moment you put h as zero. This is not a harmless factor because it will become zero the moment you put as zero, but these two are harmless factors. So pull them out. Pull them out as x into x plus zero, which is x squared. Okay. And only work with this part. Okay. This is where that, you know, zero by zero form is line. Okay. So we have to work only with this part. So let's expand this. So this becomes, by the way, x sin x plus h and x sin x, I'll take an x common. Okay. And then minus h sin x. Now this h, I will individually divide here also. I'll divide here also. Is it fine? Any questions? So this h, this h actually goes off. Okay. Now, all of you please pay attention. I'll just make your life easy. This term I can actually write it like this because x is the term devoid of any h. Okay. And this term is anyways, devoid of any h again. Now, what is this expression? Let me just write it like this. What is this expression actually? Anybody knows anybody is familiar with this expression? Isn't this the derivative of sin x with respect to x? So doesn't it not become cos x? Okay. And there you go. This is the answer for this question. x cos x minus sin x by x squared. Is it fine? Any questions? Any concerns? Clear? Okay. Let's take one more because we just want to practice it a little bit more. Let's take it from here. Can we do the third one? Can we do the third one? By first principles, even though the question doesn't say by first principles, we'll do it by first principles. Just the third one. Okay. Should we follow the first principles, this definition and solve it? For a change, I will use this definition now. x minus h by f of x by minus h. Correct? But this also fits our definition of first principles, change in the output by changing the input when the change in the input is very small. Correct? So let's use this definition as well. Okay. Let's see. f of x minus h will be one by root x minus h by one plus root x minus h. Okay. Minus f of x, which is one minus root x by one plus root x, all divided by minus h. Okay. Let's take the LCM of the numerator terms. So that'll give us one plus root x times one minus root x minus h. Minus one minus root x times one plus root x minus h, all divided by one plus root x minus h, one plus root x into minus h. Is it fine? Any questions so far? Any questions so far? Okay. So now one more thing I would like to highlight here is that these two terms are harmless. The moment h tends to zero, they just become one plus root x. So I can bring them outside the purview of our limits and write it as one plus root x the whole square. On the numerator, you'll end up getting one minus root x minus h plus root x minus root x x minus h. And the second two terms will actually give you one plus root of x minus h minus root x minus root x x minus h. Okay. All divided by minus h. Now on opening the brackets, on opening the brackets, you would realize that one will get cancelled and the last term will actually get cancelled. So you'll end up getting something like this negative two under root x minus h, correct? And two under root x whole divided by minus h. Okay. In short, you'll end up getting two by one plus root x the whole square limit h tending to zero root x minus root x minus h by h. Okay. Now this is a simple limit question to solve. Just rationalize just rationalize by multiplying and dividing with the rationalization factor. Okay. So this will become x minus square of this divided by h and you have something like this. Anything that I have missed out, please bring it to my notice. Oh, there was a minus sign also. Sorry. Yeah, minus sign a little bit. Okay. So now opening the brackets here, you will get only h and this h and this h will anyways cancel each other out and put h as zero, it will become two root x. This two and this two will also go for a toss. Is it fine? Any questions? Any concerns? Please check all the steps and let me know if I missed out anything. Immediately bring it to my notice. Okay. Let's do this question. I'll be giving one from my site. Find the derivative of this function, ln of, see, you can use any definition Siddharth. That's what I was trying to say. What definition to use where there is no such restriction. You can use any definition anywhere you want to. Just because I use this, use the three different expressions so far. That doesn't mean that I figured out that this is, this will work for this. No, it is just to show you that anything will work for the quickest result. Use the very first one f of x per sec minus f of x batch. That is the quickest one. There's nothing like time order. Okay. This will give me a faster result. This will be anything can work for anything. But yes, you have to be good in your evaluation of limits. That's it. Yeah. Let's do this. Find the derivative of this with respect to x by first principles. So today, I will not be doing anything other than the first principles because no, we don't have much time. But of course, in the next class, we'll be, you know, wrapping up this chapter. And not only that, we are also going to start with conics. Okay. Let's decide to use this expression. Okay. By the way, I'll give you some time because the most of you will be trying. So please do so. I'll start discussing after three minutes. And if possible, please give me your response as well. Okay. Should we discuss it? So limit f of x plus h will be what? f of x plus h will be ln x plus h the whole square plus 1. Correct. And f of x is ln x square plus 1 whole divided by h. Yes or no? Now, let us use our log properties. Can I say ln a minus ln b is ln a by b? Now, a few please pay attention. I'll just make some small cosmetic changes here. I'll write it as x square plus 1 plus 2x h plus h square. Okay. And individually divided by x square plus 1, x square plus 1. And of course, there's an h line. Is it fine? Any questions so far? Please immediately highlight. So this is as good as ln of 1 plus. Now, see here, this term I can write it as h 2x plus h by x square plus 1. Am I right? Any questions? Any concerns? Okay. Now, all of you try to recollect when we were doing the log limits. Okay. Just recall here. ln of 1 plus any function divided by the same function. And let's say x is tending to any value A where your function was such that the limit of that function as x tends to A was 0, then this answer used to be 1. Am I right? Do you recall this? Now, I'm trying to create a similar situation here as well. So here you have a function of h, which is tending to 0 when your h tends to 0. No doubt about it. But the problem is, I don't have the same function over here in the denominator. So here also have to create the same function. So for that, what will you do? For that, what will you do? You'll say it's simple. Multiply and divide with 2x plus h by x square plus 1. So I'm multiplying and dividing this entire expression with 2x plus h by x square plus 1. Yes or no? Am I right? Does this make sense to everybody here? If you have any doubt, please get it clarified. Okay. Now, what about this fellow? Let me just split this limit also separately. It doesn't matter because product of two functions, I can always evaluate the limits separate, separate also. So tell me what is this expression going to give you as per our standard result? This is going to give you a 1. Absolutely. What is this going to give you? When you put h as 0, 2x by x square plus 1. So overall, what's the answer to this question? This. Okay. So this is the derivative of ln of x square plus 1 with respect to x. Is that clear? So a short, short question is going to come for your school exams, UT semesters, whenever you have it based on first principles. So ensure that you have practiced at least the basic NCRT questions or RD Sharma questions based on this topic. I will be also sending some DPPs today. Okay. So please ensure you have done the practice for this. Okay. So we'll stop the session here today.