Derivatives

Ratnakar Yedal
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Calculus Made Easy

Calculus Made Easy

03:08:24 of on-demand video • Updated

  • Concept of Limits
  • Differentiation and its applications
  • Integration and its applications
English [Auto] Now let's look at differentiation in the differentiation. I'll decoding that. It is a truthful differentiation. Or use changeroom. They don't make the connections exponential functions and logarithmic functions. Now let's understand the concept of derivatives. Here we have a AROUND AND A point B on the curve to define the slope of a row at point B. We take a point Q And the code I'm the line joining the B and Q is called the second line at B and its slope is denoted by MP Q.. It can be found using the coordinates of B and the coordinates of Q And if you like Q Name along the color then the slope of the line P-Q changes and the of course supples rule is given by. Why is equal to a complex. This is the equation of the code I and F is function in B of X not f of x not as a point on the curve for any point Q On Roe v q is not equal to P. It's X Gorney can be has X not plus H made it is not equal to zero. If it is greater than zero Cubas on the right side and if x is less than zero Q will be on the left side of B The point B is coordinate will be X not. Come on F off x not. I end up going to Q's coordinates are x not plus H comma f of x not because q is close to B. I'm it is about a distance along the x axis. No the slow MP Q of the second line p q is given as the difference in the White coordinates of wine Q and B. That is all x not plus minus for X not daunted by x coordinates it is x not because edge minus x. So this can be simplified as far as not plus minus for X not ordered by a judge. Note that as Q approach must be the number edge approaches to zero from these we see that the slope RHO at B which is denoted as MP is nothing but the limit of 4 x not Plus edge minus X not guided by Edge as etched into the zero. So no we are trying to find the slope at the given point. As you can see we tried to find the velocity at a given instant of time. So similarly for trying to find the slope at point B. No let's look at an example. Find the slope of the curve going by. Y is equal to x quite at the point B they come. My question is f off X equal to x choir. And the formula for slope at point B is the limit of f.. All three plus minus F-off three girder by Edge as echange to zero at the point B has ex-colonies of 3. This is equal to 3 percent. All square as if all three plus edge means edge was minus is required by X and upon simplifying this we get nine plastic set. Plus it's quite minus 9 Good by Edge which is equal to 6 edge plus at Square by Edge which is a total of six plus such as etchant to zero. As you can see when it is zero. This is simply equal to 6. So we can get back the slope at point P is 6 No a function f is differentiable. Means that for every x boloney may not have the limit of the difference or if Bichette that means the limit or X plus minus X by it exist. So this is the slope. Suppose we are given a function you call X I and the limit of half of explicit minus apophatic by. And it's just like this. That means we can calculate the slope at any given point on 4 x then if the limit is the real number and its value depends on x. In this way we get a function called the debt we do of f.. I'm denoted by f dash from the mean and 2. Ah so this is the point of differentiation. So the differentiation here means we are arguing a function y you call the X. I think V can calculate the slope at any point on the graph and it exists. Then we call that as the degree we do f and to not get s f Nash. The graph of f is a curve. The assumption that f is differentiable function implies that at every point in the car the slope exists at the point whose escorting it is x not the slope is a flash of x. Thus f can be considered as a log function. That means that it is nothing but a slope function. So the is a limit of X percentage minus about X bite h as it's just a zero exist. I see that that he would do X. Let's take an example. Let's get it. And then he would do all f off X which is equal to x quite to find a way to have means to find that Tamino bash. I'm finding out what I shall fix now. Using the formula for that it would do that it would do. Means Exley bit off half off as such. Minus half off x. Go read about it as it's changed to zero then the limit of X plus H is nothing but it's just a total square minus f off xes Esq.. Altered by edge when expanding we get X quite close to x x plus it's square minus X quite altered by Edge and simplifying we get to exit. Plus it's got all that about edge and this is called the limit of X plus it has its roots to zero and it is zero. This is just simply so that there is no cure. It's quite easy to look at. Now let's look at the terminology. The process of finding that they were to use is called differentiation. In a previous example to find the slope of the parable of what is equal to x which is equal to x quad at point 9 we use the definition to find that right. So we used we calculated f dash of f off explicit. Normally the process of finding that that he would do. It's called projection. In the previous example to find the slope of the bottom block. Why do you call that squired at point three come on might we use definition to find f t. In fact if we know the back f natural lexical to X then by simply substituting X's call to 3 we can kind of play the slope of the curve at point 3. That is f dash of 3 is equal to 6. It is very easy so we should know that that he would do the function. Then we can directly calculate the slope in the next section. We discussed how to find actual facts using the rule of differentiation F natural FX not E's called that he would do all f at X to represent a function. We sometimes write white Autocar eggs. Similarly the function can be represented in many ways as Ephor dash or white dash. You labor DX be a B-boy. F. Nash effects on the body itself affects following one of the examples we need to try to find the national export functions is trying to solve these examples to understand this better.