![]() So we have nine divided by three, that's three. But when we take it out it comes three so we take the square of that becomes three. So we can simplify this further and then nominally four X. of maybe four x squared last 36 overnight. And then The nine is outside the denominator. We would get So this becomes eight becomes 94. What? So we simplify this a little bit further. Yeah, the nine is also yeah, 98 of the nine X. So let's see why prime is equal to So we have one half times for a square 36 overnight. And now we want to take the derivative of this. Okay, so we can keep yes, so we could simplify it but we will keep this for now. So what I say called uh square root native for Have squared last 36 overnight. So we have negative for X squared minus 36 plus two. So the equation we have here, we move everything Mhm to the right to get right myself the same. So paul beat, we want to write why in terms of X. Okay, so We can simplify this by divide by two. This means that by prime is equal to negative eight X. However, the second method is definitely the easiest one to use, so it's always good to look for simplifications that you can make the original function before you apply these rules. Therefore, they both give the same answer. Where is the other one? Took a bit of simplifying and rearranging. ![]() However, one of them was just a single step. And if you take the derivative of that, that is why prime you'll find an equal to root of X is one minus Trudeau of a zero. Factor out an X minus A from top and bottom, leaving us with just X minus A. So our original function Justus Reminder, is y equals X squared minus a squared over X minus. Now let's try the other method of getting the answer. All right, so that was a decent amount of work, but it gave us an incredibly simple answer. Okay, that's the same term on top and bottom, meaning everything is going to cancel, leaving us with just one. So you may notice that on the top we have this two x squared and minus X squared so we can perform that's attraction to give us just X squared minus two a X plus a squared divided by X squared minus two, a X plus a squared. So I plus a squared because the negatives will cancel. Eso We're going to have X squared minus two a X minus a script. The negatives cancel right here and then this is divided by Well, let's expand this out. So on the top we can distribute out this product giving us two X squared minus to a X minus, X squared, plus a squared. So driven X is just one and derivative of a is zero divided by the bottom term squared X minus a square. So x squared minus a squared times derivative of the bottom term. X minus a times derivative of the top term two X and then a is a Constance that just goes away minus the top term. So why prime using the quotient rule is going to be the bottom term. So in this case, axes are variable on A is just a constant. The first way we're going to try is just brute force quotient rule that is, we're going to apply the quotient rule without simplifying at all. Why, in two different ways, let's get started. We're going to find the derivative of this function.
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