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Complete the proof of Note by showing that $f_{y}=Q(x, y)$....

Question

Complete the proof of Note by showing that $f_{y}=Q(x, y)$.

Complete the proof of Note by showing that $f_{y}=Q(x, y)$.



Answers

Prove using the definition of derivative, that if $ f(x) = $ cos $ x, $ then $ f'(x) = - $ sin $ x. $

Shin, and that's in vain. So why, even in verse, this implies that why applied at inverse over your toe? And it's the same thing as saying that actually inverse of Okay, So the fact is that every function of India that that he could do injury and it's

In this question, we're going to compute the partial, the relative of you, and check if you use a bath way. Go to use a word X. So first we need to compute just two terms and we compare the use of X first. So one little director of U with respect to act, we treat why, as a as a constant so science saying Why is also constant. So it's basically the same. Y is a coefficient of e to the X y. So we're just right in here and then we need to take the relative this part because it'll the XY contains X. We need to use the chair rule, which is the directive of the whole thing. Multiplies it by the directive of the exponent, which is why so it will become something like this. And then we take the right to vote you with respect to y. For this one, we need to use the product rule because these two terms there are they both contain wise, which means there are two functions of why multiplied Tizer. So we use the product rule and he says the product. Sorry. The directive of the first function multiplied by the second function and the first function multiplied by the director with the second function. So the same way. Brian, Where's the cool saying white? Right. Okay, so this is a directive of YouTube. That's why with respect Why all right, so then let's compute this use of wax that waited directive of you of why, with respect to X. So let's see the first term. So the first term these two contains X right? This one is is a constant. So we write that down and for this to we need to use the product rule again. So it will become just and remember that the we need to keep the coefficient for both of these terms. And for the second there, there is only one term can hence x, right. So this part is basically it's a coefficient, and we just were there long and take the right towards this part with respect to acts regardless. And if we simplify that, we will get something like this. And now we did a relative of you suburbs with respect to why, um, this is a little bit complicated because there are three functions of why all of them can. Hence why? Which means we need to use the product rule twice. So let's just consider that these two, um, constitute one function. So by using the product rule, the primary the first function multiplied by the second plus the first function multiplied by the the relative of the second right. And now, then we need to compute the directive of these the production of these two terms. That means we need to use the product rule again to get this and then if we can qualify, that we will get something like this. See, these two are exactly the same fighter e to the X Y Out in the bracket is a sign y multiply x y plus one. We got one here and another term is closing. Why multiplied but white? We also get that. So this is how to prove that you survived S y equal to use a Y X

So we're told that affects is equal to G of axe is equal to act. So both of these functions are just X on. We're going to use this to show that F prime over G prime does not equal f over Jean Prime on DH. This is basically showing us why the kosher rule is important of I know the temptation is just to take the route of the numerator over the druid of the denominator. But that's not how it works. That's not going to give you the right answer. So first, let's do it this way. So when we have f prime over g pride f the vacs Oh, Virgie of ax is XO brax. So this first way take the purity of Axel that's just one Trudeau backs again is one one divided by wood was equal to one. So we would think that this is the derivative of this is just one. But let's do it the correct way. So the correct way is using the quotient rule. So we take the drawer. The first function times A second bite is the root of the second function times. The first over the second function squared so derivative of X is one and we rewrite our second function Times X minus the druid of the second function, which is one times the first function, which is X. Nothing on DH. This is divided by the second function squares, a divided by X squared and in the numerator Here we have acts minus X over X squared and that's zero over X squared and zero over X squared is just zero on, and that's a real answer. Now, if you look here except Max, that's just equal to one. Ah, the derivative of 10 We know that, but here in this first way, we we take the route of the top divided by the troops of the bottom. We get one as our derivative, which is not correct on. And so one is not equal to zero. And that's showing that this right here is not true. So f prime of axe overdue problem next does not equal f over g pride

In the question. We have to find the indicated higher order partial derivative. It is given effects by for that equals toe Helen Off X minus y now moving towards the solution effects. Why would be cool toe del by del y off del by Dell Eggs Ellen X minus white which is a con sto del off del mar del by del y one upon X minus Y dealt by Delta X x minus Why it will be equal toe del by del Y one upon next minus y in tow one minus zero. Now it is because create why, as if it were a constant while you are differentiating with respect to X so that by delve I X minus y to the power minus one, which will be quite toe minus off X minus y to the power minus 20 minus one that is equals to one upon X minus y to the power to on this would be the answer to the given question. Thank you.


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