 Then what can be function? Yes, it is differentiable at x equal to x. Same would work for difference as well. So I am writing it in one go. Can I say the same for product? Product will also be differentiable at x equal to x. Now these properties actually will make our life very simple in comparatively clear questions because we don't have to do those differentiation for example in the previous questions. We don't have to sit and do the differentiation in order to know that my left hand derivative and my right hand derivative are equal or not. And same would work even for quotient of these two functions provided your G of A is not 0. Is that right? This property you will find in almost every book. But what they don't discuss are the properties which are going to come next when a function is differentiable at A and another is not a differentiable at A then what happens? Let's discuss that. Does this need any debate or a discussion? I don't think so. It's very obvious. G of x is not differentiable at x equal to A. Then what can you comment about the sum of f of x and g of x? Will it be differentiable or will it be never differentiable at x equal to A? Or I can't say it won't be differentiable at x. I'm just saying it will definitely not be differentiable at x equal to A. Not necessarily. Then give me an example where a differentiable and a non-differentiable gets you differentiable. So what we just need is a question like x plus 1 into what A I am. What's the difference between x plus 1 and x plus 1? It's a plus not multiplication. So it's not differentiable. Sir, there is a case for x. For example, if there is a king and there is a non-differentiable, there has to be a king somewhere. No, no, no. You take back yours. You retract your message. I will try to not talk to you. It is not differentiable. What is the case for x equal to 0? x equal to 0. Okay, it's very obvious because, correct? Yes, sir. Answer the hint. It is differentiable at 0. So here, we cannot say anything. So for putting a differentiable product of a differentiable and a non-differentiable at that very point x equal to A. Where a differentiable function at x equal to A multiplies to a non-differentiable function at x equal to A to produce a non-differentiable function at x equal to A. I will preach it to you. I will bring a constant in hand. A previous function at x equal to A. Sir, where did it come from? Tell me an example of a function which is differentiable at x equal to A. Another function which is not differentiable at x equal to A. And when you multiply it, you get a non-differentiable at x equal to A. So mod x minus 2 into sin x. Mod x minus 2 into sin x, which point will be? 2. At 2. Can you please check in the graph? It has to be correct. He said mod x minus 2 into sin x. So sin x is a differentiable at 2. Mod x minus 2 is not differentiable at x equal to 2. And when you multiply it, you will get a... I have tried to see whether that claim is correct or not. Sex, let's say. We know it's differentiable at pi. No, it's not differentiable at pi. Which is differentiable at pi. Take it on its way, sir. Is this claim correct? If these two functions are multiplied, this was differentiable. Is this correct? Is it differentiable at pi by 2? I multiplied these two. Are you getting this point? In fact, we have a... You know, the whole is because of the fact that the functions making that sin x did not exist. Are you getting it? And this is the fundamental which were used to make this function. They just take it on its face value. Okay, sin x. Continuous everywhere. Fringe everywhere. Done. But the moment... No, no, no. Gentle, sir. The moment I say it's in two tan x, because pi by 2 is not in the domain of tan x, it will also be not in the domain of sin x. And if some point is not in the domain, how can you talk about differentiability there because it will be discontinuous there. And hence it will not be differentiable there. So sir, this statement, is it correct? No. Be careful of these small, small, conceptual testings that can be asked to you. Six. Or seven. Six. What's the number of two times the value of one is the mean? Yes. But tan doesn't have. What do you mean by that? What do you mean by that? The domain of the resultant function would be an overlap of the domain of the two functions making it. Not only in case of multiplication, but also in the case of sum and also in the case of division. Individually extrude extra points because you do not allow your denominator to become zero. Remember in the domain chapter last year, I talked about this concept. If I want to find domain of f of x by g of x, domain of f of x by g of x would be nothing but, let's say the domain of this is d1, domain of this is d2. What is the d1 intersection d2 excluding those points where g of x is becoming zero? This I did last year with you. So not only it is becoming zero. Because if you see at g of x, there is no harm in it becoming zero. So those points will also come in its own way. But overall in this scenario, those points should not be entertaining. Makes sense. How do you write this complete thing in one book? It's on your page. I don't know. Not differentiable x equal to a and g of x is also not differentiable at x equal to u. What can we comment on the sum of these two functions? What does that matter? Not differentiable. Everybody? One voice? Not differentiable. I have a scenario for you. We know it's not differentiable. All of you please. I have seen some t-shirts in that. It looks like this. Also there are scratches and all that. So do you see? There is a stress on the one zero. So it is not differentiable there. Let me take another function. Now you are saying can't say. When you add it, please quote this. You might be like it. That is differentiable. And this function is like c equal to zero. So do you take back your claim that what's the model of the story? Can't say. Can't say. Now why I am giving you all these cases is because many a time such questions come and people do non-differentiable functions, they are always added to give you non-differentiable functions. No. Cannot say. It depends upon the question itself. And the very same function if you add them, psi mod x plus mod x if you do, you see that it is non-differentiable at zero. Not the graph, see there is a sharp corner. Right? What about from bottom like this will come? What about their multiplication? What about their product? No way. Can't say. Can't say. Can't say. Can't say. Can't say. Can't say. Can't say. Can't say. All these if you have an installed desktop set on your phones. Facebook app you will download like this. Can't say. Can't say. Can't say. Can't say. Can't say. Can't say. Can't say. Can't say. Can't say. Can't say. Huh? Psi mod x into mod x. Correct. Oh wow. Other example would be this. I always think of this example. Let's say I have one when x is greater than or equal to zero. Minus my x is less than zero. Correct. Clearly if I choose another function, minus two when x is greater than zero. This is also discontinuous at zero. What happens when that's minus two? Five. That's plus two. Plus two is always continuous in the future. Okay. And there is no domain issue also. Please take the domain issue. It should not be like it should be the all up and down. Is it right? So cannot say and same would be also applicable if you are taking the quotient of two functions. We cannot say anything. So one question I have to ask. Is the derivative of a continuous function always continuous? Let's say a function. Continuous at x equal to eight. With the derivative. No. Is the derivative. What? Not necessarily. Not necessarily. So please remember all these things. So now we take questions.