You can't do the derivative of this function "nomrally"

[00:00:00]Today let's talk about how to find the derivative of this function at x= z. As we can see we have f ofx is equal to e to the -1 /x^2. If x is not equal to zero and if x is equal to zero we get zero.

[00:00:16]We want to find the derivative and we have learned all the derivative rules.

[00:00:19]Maybe we can just do the chain rule here. Let's give it a try. frime of x e to the something first that repeats and then I have to use the chain rule the derivative of that is the same as taking the derivative of -x to the -2 power okay and then put the power to the front minus one negative * negative we get positive2 and then x to the -3 so we multiply by that and that should do it. And perhaps I can rewrite it. We have two on the top. And maybe let me keep this on the top as well. And now we'll put only this on the bottom. I could also have put this on the bottom. But anyways, now let's plug in zero into the x. Can we do that?

[00:01:12]No, because we'll be dividing by zero in the exponent here. And another no because we have a zero here either. So what do we do though?

[00:01:22]So, as you can see, if you just do the rule, it's not going to work. And in addition, what about this thing though? What did we do with this? It seems that we didn't use it. So, could I have just replaced the zero with I say two or 17?

[00:01:38]I don't know. Well, here's the point.

[00:01:43]Sometimes if the derivative rule does not work maybe we have to go back to the fundamental and that is the definition of the derivative. Okay. So let me write that down right here. f prime of zero and there are two versions for us to use. One is when x is approaching this number or the other one is h approaching zero. I personally like to use x approaching whatever this number is. So x approaching zero and the definition is f ofx minus f of 0 over x - 0 just like this. Now check this out.

[00:02:28]Firstly we have the limit as x approaching zero.

[00:02:34]This right here minus f of zero. When x is zero the function gives us zero. So this thing here is minus zero. Good.

[00:02:45]What do we prove for f ofx though? Here remember when x is approaching zero we don't want x to be exactly equal to zero. So when x is not exactly equal to zero the function is this. I will put down e to the -1 over x^2 and then the bottom is just x - 0 which is x. Doesn't matter for now. Let me just write it down. And of course the minus 0 doesn't matter.

[00:03:10]So we are going to look at this limit and now if we plug in zero into all the axis the bottom will give us zero and the top right here is -1 / 0 squar which is negative infinity when we do limit e to the negative infinity is zero. So this is a zero for zero situation. That means we are supposed to be able to use the laptops rule but don't do it because if you differentiate the bottom you get one. If you differentiate the top as we saw earlier, we get this times 2 over x cub that expression I think it's harder than this one right here. So no. All right, I'm stuck. So why do two? Now if sometimes these things don't work, maybe do a change of variable. That will actually work out really well. Here's the deal.

[00:04:07]I'm actually going to consider two cases. The first one is the limit as x approaching zero plus and uh because there's no plus or minus. We have to consider two cases. So let's look at this one right here first.

[00:04:22]And what we are going to do is we'll do a change of variable. Let's say t equal 1 /x.

[00:04:31]And as you can see that means okay let's just keep it like this we get the limit as x approaching 0 plus 1 / 0 plus will give us infinity and then you will see that we have 1 /x which is the t right here and then this is going to be e to the negative and then we have 1 /x² if you square both sides That's just t right. So t ^2 is equal to 1 /x^2. Yeah.

[00:05:09]So now this is actually better because if I bring this down to the bottom, we get the limit as t going to infinity.

[00:05:17]The top is t the bottom is e to the t².

[00:05:23]We get infinity over infinity situation.

[00:05:26]I will do laps rule here.

[00:05:29]ddt ddt.

[00:05:34]So we will get the top is just going to be one and the bottom we get e to the t² and then use the chain rule here we have to multiply by 2t. Now as t goes to infinity on the top is just one on the bottom is infinity. So we get to draw a conclusion and we see this portion right here will give us zero.

[00:06:00]So that's good.

[00:06:03]And now for the other direction I will just put down similarly pretty much the same thing and you can work that out on your own when you have x approaching zero minus be careful with the negative and all that stuff but you will also end up this being equal to zero as well. So all in all we can come back here and say that this limit is equal to zero.

[00:06:30]So that means our derivative here is just equal to zero. So this right here is a correct solution for this question.

[00:06:39]But I'm not sure how I feel about this question though because I can think about so many wrong ways of doing this and then get the correct answer because the answer is just simply zero. For example, somebody could just differentiate the the first piece which is 2 e -1 /x^2 over x cub and then rewrite if x is not equal to zero and they can differentiate zero which is zero and then maintain if x is equal to zero and we want frime of zero. So they say by looking at this we should be done.

[00:07:23]No not correct.

[00:07:27]not correct because the danger part is if you have this function let's say we change the zero to let's say five or 17 you guys know I like the number 17 if you differentiate 17 you get zero so this right here is still true and uh is still zero though not anymore the reason is because if you change this to 17. The truth is this function here is no longer continuous at x= zero. If the function is not continuous, you are not going to have derivative at all. So the derivative will have been d and e does not exist. So be really careful.

[00:08:17]Yeah, that's it. And let me just change back the 17 to zero. I think.