 Yeah, so when we say D by D, okay This quantity at 9 30 p.m Now, how would you address this question? You'll say okay exactly at 9 30 point zero zero one second How much did my car move? Divided point point zero zero zero one second, right? That would be the distance also be very small So this ratio those quantities and thereby limit is helping us But in this there's no displacement in the car So even if you take the displacement is 0 gf or gi a function if I'm trying to find out the right So when you say gi of minus 2 by h It is actually 2 minus 2 by h which is exactly 0 Correct you see the graph This is a graph. Yes, sir minus 2 by negative h Correct Now we'll tell you what is gi of minus 2 Correct divided by minus x which is actually 1 by h So can I say this is a 10 to 2 this line will have in five night So that will almost be vertical to the Ice axis For that matter any integer points with respect to gi a function My function will not have a unique tangent In this case, I'm getting the fact that there cannot be a unique time Yes or no proofing So there's no unique tangent That means from the very basic definition that I wrote it must have a unique tangent with the finite slope That is null and void So we say that the function will not be differentiable. So here's the rule that I want is continuous At x equal to a Not differentiable at x equal to a Let's check the continuity first before we check the Differentiability of a function at x equal to a So make it a habit whenever a question comes to you about Discussing the differentiability Always the root goes via So first Continuity test We'll do the continuity test first and then we'll go to the differentiability test Okay, but in this case you must also label It must be continuous But if the function is Discontinuous it will definitely be not differentiable So here very important thing that you should note now If Is differentiable At x equal to a Implies f is definitely continuous at x equal to a is continuous at x equal to a at x equal to a It ensures continuity, but does continuity ensure the sensitivity? No But if you are able to figure out the point x equal to a Differentiable Is that right? Will not be differentiable Let's take the graph Model x graph is a v-shaped graph like this, correct? Differentiable at zero, but hence we have to do the differentiability test Now h that means we're coming left to x point, isn't it? So in this case, what will i do minus h minus f of zero by negative h? This is what i have to do right Yes, take h is always on the full wave of zero graph So even when you do a zero plus h Which will again be your h zero positive So for that h has to be positive But just by stating the fact that h is tending to zero Can you say that it's because they think that h is already treated as a positive quantity very small positive quantity But when you say h is tending to zero it doesn't actually imply that Yes, so we're going to be more accurate at the rate of x tending to zero plus f of x minus f of zero by x minus x minus, i mean i'm going by the general General way of expressing Yes, extending to a plus f of x minus f of a by x minus So that's more you think In the end when you express x is equal to x minus is equal to Yes, when you say x is a plus h but h has to be a positive quantity Anyways, this will not affect your calculation though. Don't be scared that we have to put an extra expression over here now All of you mod x function mod x function is defined as x When x is greater than equal to zero and negative x when x is within zero, right? Right So if i ask you what are the assessments you're slightly less than zero What would the it'll give you minus of whatever input you have made through it, right? So you have made an input of negative x to it Negative of negative x will become positive x So you can actually say zero What is negative x negative x so my answer is one You know, what does it signify? It actually signifies It actually signifies The slope only left of zero See here you actually chose a point zero minus Yes or no And like yes, and where is that tangent line will have a slope of negative one Right because angle of inclination is 135 degrees, correct? Yes or no? Any question? Any question? Okay On the other is the shortcut that we normally get So Zero Minus It gives me one as the answer Signifying the fact that if I take a right of zero and construct a tangent You can see the pink tangent that has a slope of plus one So if two soaps means two times so the unique criteria is violated Right, remember right So here ought to be differentiable at this point Now graphically, sorry D and RHD in this case So we'll lift your function will exhibit a kind So we're not differentiable at such points and of course when you see discontinuity also Right So if you ask differentiability mean to him, you will say it's the absence of discontinuity It's the absence of any corner. It's the absence of any cusp It's the absence of any king on the graph at that point Right as a layman, right? So you remember there's a discontinuity Is that funny? Can you give me some examples where a function is discontinuous at a point because of at least two examples What x is something which I'll give you a part of this give me two more examples Mod of log x Mod of log x which point you should all will mention the point At x equal to one Mod of log x at x equal to one so he says if I Yes or no Earlier it was it got reflected up. So what is getting formed over here? A kind of in this case at x equal to one this function will not be differentiable Correct another x equal to one and this is at x equal to zero at x equal to zero You can clearly see the cusp getting formed at these points cusp If you talk about points, Siddharth is saying if I'm not wrong He's saying that It will show a corner at that point for example mod x Right mod x was becoming zero At zero It gets reflected on differentiability point at zero Sin x sin x was becoming zero at zero. It was becoming zero at pi. It was becoming zero at two pi It was becoming zero at negative pi the moment you want differentiability will add up Mod of cross x If mod y instead of x it's going to become undifferentiable at certain points along the y axis Or it depends on the function depending on what the function is At zero Where is it becoming zero When you mod it at those points where the function was becoming zero it will not be differentiable for sure mod x plus two Is it differentiable at minus two? No You will always Is that is the idea here? It's not showing discontinuity It is not showing kink. It is not showing kormel At x equal to eight infinite so Then even at this point f of are you getting this point? That means F dash a will not exist There are many functions which can show the graph like this. In fact, uh, what exists? Can I say? So sin inverse x will not be differentiable Differentiate sin inverse x we used to get one by yes or no Thereby the function will not be differentiable at one Okay, this can be treated as a very good example Is that fact my function will not be differentiable. So ultimately when you say a function Differentiable at x equal to eight. What's the final condition that I should keep in mind as a student who is going to like school exams Yes L hd Is equal to self is called f dash a when these two are equal it itself is called f dash a and his f dash a will exist If these two are not equal or if these two are one of them is not finite or infinite We will not have plain hands here Okay, okay, so let's talk the function is continuous Which means it is continuous at the interval or the point Never state these things without an interval or a point Right, then we'll have is differentiable or f dash exists in an interval Open it to be finite Okay This is simple Slope should be you learn both half of it half of it work The hypothesis is correct, but the way you are going to be should be finite Please And for all the points it is sufficient for us to show that it has a finite slope at the bottom points That means the right hand derivative a should be finite He should be finite. In general, I would be equal to a finite value. This is the whole chapter all about Like that chapter will keep it simple. You will remember it for life The more you complicate it the sooner you'll forget it Right, very little organ. It doesn't like to remember anything which paints it That's right. The more we do exams and start Enjoying things that will