Always consider this function. It affects T. Which is uh that -2 Squared of T over one plus. So I want to find those derivative. I mean just like the last two terror before this one, we're just gonna consider anything that is not X as a constant and anything that has an X. As a variable. So this one is just gonna be, you know, square root of X can be written as excellent power one half. And the derivative is going to be one half excellent, negative one half. So that is a derivative of any squared effects. So this one is going to be just X to the negative 1/2 because this is the derivative of the screwed effects. And then whenever you multiply by two, it just takes away this one half. Right? So this is what you have then minus. Oh, I beg your pardon? Uh I'm going to use a coaching role. Okay, I got carried away by this derivative thing. So it's just gonna be one plus. In fact we're not going to use a caution rule because the denominator here does not have a annex. So it's just a constant. Right? So what we're gonna do here is separate. This this uh fraction. We're going to do This over one plus The -2 T. Over that. Yes, this one is a total constant because it does not contain an X. So this one's derivative, this is going to go to zero. Yeah, I mean that is still this right? This this one is still this one. I just separated it. Okay, so in the eyes in the context of uh partial derivative of the function of respect to X. This one does not contain an X. Does not have any X. Here. So it's just a constant. So the derivative is going to go to zero. So we're just going to concentrate on this one. And I, like I said, the numerator contains the next and its derivative is excellent. Negative one half over one plus. Right? Because uh square root of X has this Esther as the derivative. And I want to really multiply by two. You have justice. Right? So this is what you have. Another person can want to put it in this form. Yeah. Right, That is entirely up to you, right? Or you can make it this way. Again, one over square root of X plus two squared of X. T. There are many forms you can put it right, This is correct. This is correct. This is correct. Right? Any which way? No, I want to find a partial derivative with respect to uh T Okay, now, since the denominator contains a T, it would not be too wise to split it up like that. You can just consider it in this compact form. So since the denominator has T. We can now use a question rule. So the question rule has says to have the denominator squared. So this is gonna be one plus two squared of T. Square right? Because the denominator contains city and we're taking partial to a bit of a respected T. And so loaded high is going to be one Plus 2 Description of T. Now this one is saying is completely a constant. So it's going to go to zero what is to part of what is a derivative of this one? Well you know this squared of T. Behaves like this thing as soon as I'm gonna put team place effects. So whenever I multiply by negative two all that I have is negative T. To the negative one half. That's all I have then minus loaded. High. Hi. Hello actually so you maintain the numerator and then you differentiate the denominator right? So that is also gonna be T to the 1/2. Right? So you got to clean it up a little bit. So this is gonna be okay. Uh You know negative one plus That teach the negative 1/2 in minus. Uh You know Now this teacher 1/2 is the same as the square root of tea. So I'm just going to have XT right? Because this square root and this is square root of X. So it's just gonna be squared of X. T. Then minus plus because this is a negative. So plus do now this is screwed of tea. Time squared of tea is actually gonna be T. Right. You have a script of T. Times square root of tea is the same as squaring of T. Square. And the square council cancels the the square root and you have t. Okay that is what is happening. So you have this one then over The denominator which is one plus. That's okay. So uh yeah you can pretty much leave it like this. This is not square root of tea because it is not t to the power one half. It is t to the power negative one half. So it's basically going to be this over that. Okay, do not get confused. T to the power of one half is square root of T T to the power of negative one half is one over the square root of T. They're not the same. Okay. That is why I couldn't take this one to foil. Right? Because technically is going to be under is going to be over square root of tea. That is what this one means. Okay so you can just multiply Yeah uh but what you can do is I mean whatever you foil, you're going to have negative t to the negative one half minus two because this is teaching at one half and this is teaching the negative one half. So whatever you multiply, you're gonna have just to to the zero which is just one in minus XD plus two T. Okay. Yeah, so basically this is what you have and the notably one plus two squared of T uh squared. Ok, so you have this one? Yeah, you can make it, aye, aye sir or you can live it this way, any which way is fine. Let me ride those two here. Uh You know whatever you try to make it nicer Yeah, let me I want to make it nicer. A little more compact and easy to understand this thing right here. Okay, is the same as one over square root of two? Like I said, so one over square root of T. What I'm gonna do is bring this one to the bottom so I bring it here. But when I do that then I got to do a few things and I got to multiply each and every term here by square root of T. So this is gonna be negative. Okay. Yeah negative one because there's gonna be this negative one on top. I just brought this script of teacher denominator. So I have negative one. But here is gonna be negative two squared of tea. Right? That is the small thing I gotta do when I bring this thing to the denominator. And then this thing is gonna be negative primitive X. Because yeah I want to write multiply uh this by screwed of T. You're gonna have square root of X. Squared of tea time squared of T. Multiplying this thing by square root of T. And this spirit of T. And the spirit of T. Is going to be T. Actually. So uh what's you gonna have is justice this tea? Okay then. Plus two. Whenever I multiply this one by squared of T. I get T. To the three of the two. Right? Uh There's a little mistake. This is supposed to be negative one half, you know? Uh Yeah let's go through it again. This is protecting the partial derivative respected T. Okay so we're gonna use a caution rule. The question rule is going to have this denominator here as squared. So that's what you have here. Now the numerator is going to be. You maintain this one as against taking the derivative of this one. So that is what you have here. And the derivative of the numerator is just negative T. To the one half. And then minus you maintain the numerator and to the review of the denominator. So the numerator is this guy. And then the derivative of the denominator is supposed to be negative T. T. to the negative one half. Right? Because the denominator here is going to be the derivative of the derivative of this. Denominator is just gonna be T to the negative one half. Because chest this one is one half T to the negative one half. And I'm multiplied by two. Then it takes away the so it's T to the negative one half. Okay, so that is a little correction I wanted to make here. So yeah that is an important development. So then we just have to uh Revised this one a little bit. So this is going to be okay, You know, two square root of X -2 squared of T. Uh Let me put it this way, one over square root of tea. This tea to the negative one half is the same as one over square root of T. So this is gonna be the same thing. One of the spirit of tea. Okay, so now I can safely bring this thing this script of T to the denominator without having to make any fuss. So when I bring it to the denominator then the numerator is gonna stay the same. The numerator is gonna be negative one plus square root of T. Two squared of T. And then minus. Okay I can bring the denominator down because this and this are common. So I can just factor it out and then bring it down. That is why I have this one. Now the new, what is left is gonna be just this portion and just this portion right, descriptive. These now at the bottom here and now I'm going to foil right when I foil this is going to be negative one minus two when I'm for this one is gonna be negative two to the X. Plus three to the Uh two times screwed of T. Right? So two times creative teeth negative negative is gonna foil. So you're gonna distribute this negative to these two terms and you're gonna get this right and then you distribute in this negative to these two terms as well. So you have this one no negative to to the script to times square root of T. Plus The positive one is going to go to zero right? It's like having two times creative team minus two times creative T. So you just have negative one minus two squared effects and over screwed of T. One plus twos. Yeah, this is what you have. So that is the partial derivative with respect to T.