How to differentiate a fraction (function). Differentiation of y with respect to x.

Welcome to my channel. Today we are asked to differentiate this particular equation with respect towards x. Now there are two ways to do this. First, I'm going to introduce you to the Shortest way. So watch me and see how I'm going to do it. We are given something like this. The The first thing you need to do is this: if I should do this like this, that's why y is equal to 3x to the power of 4 divided by 2x squared plus 2x squared divided by 2x squared minus 1 divided by 2x squared. I haven't really changed anything. Yes, I haven't, and that's because this 2x are squaring, which means dividing each of these numbers that we have as the numerator. Now what do we do? We Go ahead and solve this, and y is equal to I'm going to take you to indices and the law of indices We are told that x raised to the power of a divided by x raised to the power of b equals x; what do we pick for one x? That is, you pick one base; it must not be x, or any base at all—once they are of the same base. Pick one of the bases and then subtract the exponents; that is, a minus b, a minus b. Well, luckily For us, what we have here is x as well, so when it comes to this place, 3 divided by 2 is going to be 3 over 2 x to the power of 4 divided by x to the power of 2 x to the power of 4 over x the power of 2, if we should make use of this law, we are going to have x; we pick one base and we subtract the power 4 minus 2 means you are going to have x to the power of 2, so when we now x to the power of 4 divided by x to the power of 2 will leave us with x squared remaining plus This divided by this will give us 1 minus 1 over 2. Now there is something I want you to know. This particular one, we have 1 over 2 x squared; you know, it's the same thing as 1 over 2 times what 1 over x squared, these two are the same. Now, 1 over x squared is the same thing; that means you compatibly you can still write this as 1 over 2x to the power of minus 2, how this is also indices when you have 1 over something, for example, 1 over 3. Okay, if you want to, okay, let me still use x: this 1 over x. square it, if you want to remove this 1 over, okay, you're going to replace it with negative, so it's It's not going to be x to the power of minus 2, that's because you removed 1 over, that means I'm going to replace 1 over x squared here with x to the power of minus 2, okay, so we go ahead and differentiate This is now easier to differentiate this from this, so let's do that now in differentiation for those. that don't really know how to differentiate, it's actually very easy. For example, if you have 2x square it if you are asked to differentiate this with respect to x, that is, if this is y, you are asked to differentiate this with respect to x, you are going to do this times this—that is the power times the coefficient, so it's going to be 2 times 2. Now you write your x squared, but you now subtract. 1 from the power, okay? You write 2 times 2, which gives you 4, right? Then write your x squared, but subtract 1 from the power, so that is exactly what we are going to do here, so we are going to_placement_ Differentiate this particular function now. Let's go ahead and do so. So I thought that, let's now do. our differentiation, that is, the y over the x, so the y over the x is going to be equal to we are We are going to multiply 3 over 2 by the power, so we are going to do 3 over 2 times 2 over 1. is the first thing we need to do, then we are going to write down what this thing is, what we have here we're still going to write our x to the power of 2, x to the power of 2, but we are going to subtract 1, just as I explained, now when you differentiate a constant, you are going to get 0. Now, 1 is a Constant 1 is a constant; it does not have x. Notice that this one has x, this one has x, but this is just 1. A constant, it doesn't have x; now, because it doesn't have x, it is a constant; it doesn't have y. Also, it's just an ordinary number; when you differentiate a constant, you get 0, so when I differentiate this one: I'm going to have 0 now; minus, I will now go ahead and differentiate So, this particular term here—how do I do it as usual? This minus 2 is going to multiply 1 over 2. So you have minus 2 times 1 over 2, then you write your x to the power of minus 2 again. x to the power of minus 2, but this time you subtract 1 from the power—minus 1—that's all Yeah, very easy. That's all you need to do. Okay, this times this, then that is minus 2 times half. you write x to the power of minus 2, x to the power of minus 2, but now you subtract 1 minus you now subtract 1 from the power, excuse me, so that is why you have minus 2 minus 1 there So we are done with the differentiation; now we proceed to solve what we have inside the brackets. So y over x equals these two, which we can see to your left: 3, then x squared minus 1. I will give you 1 now you will agree with me that 3 that x to the power of 1 is still x so it is Not mathematically wrong, right? Sorry for you to write that one now. Minus, I'm. I'm going to ignore this: zero minus times minus is plus, so I'm going to write my plus without wasting Time two, we can see I will have one remaining, right? So that means I have one x here, so no need. Of writing one x, I'll just write my x now minus 2 minus 1, I'll have minus 3. You can see how easy. It is now, if this is the first time you're coming across my channel, please do well to give it a Okay, just subscribe to the channel and click the notification bell so that you'll be notified. Whenever I upload a new video, please help me by commenting, liking, and sharing so that this The video will be distributed to those students who need it for their exam. All right, there's another way you can write this is that this particular solution can still be written as 3x plus Remember that when we did 1 over x squared to eliminate one over x, we replaced one over x with x. with a negative right, the same thing—if we want to return this back to this, we are going to introduce One over to remove the negative, so this is going to be another solution. Thank you so much for Have a lovely day, bye