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C4 Implicit Equation

Hi,

So I have tried the following question a number of times and continue to fail to get the answer, I think this is because of an incorrect rewriting of √(xy).

So the question is:

Find an expression in terms of x and y for dy/dx, given that:
√(xy) + x + y^2 = 0

Firstly could someone tell me the correct expansion of √(xy) as I am writing it as (xy)^1/2 and getting confused from there. So if someone could write this as x^ay^b instead I feel that that would help a lot (:

P.S Wasn't sure how to write it so x^a means x to the power of a...
Reply 1
Avatar for davros
davros
16
Original post by BryonyG98
Hi,

So I have tried the following question a number of times and continue to fail to get the answer, I think this is because of an incorrect rewriting of √(xy).

So the question is:

Find an expression in terms of x and y for dy/dx, given that:
√(xy) + x + y^2 = 0

Firstly could someone tell me the correct expansion of √(xy) as I am writing it as (xy)^1/2 and getting confused from there. So if someone could write this as x^ay^b instead I feel that that would help a lot (:

P.S Wasn't sure how to write it so x^a means x to the power of a...


There's nothing really subtle going on here:

(xy)1/2=x1/2×y1/2(xy)^{1/2} = x^{1/2} \times y^{1/2}

as with normal products of powers :smile:
Reply 2
Avatar for BryonyG98
BryonyG98
OP
3
Original post by davros
There's nothing really subtle going on here:

(xy)1/2=x1/2×y1/2(xy)^{1/2} = x^{1/2} \times y^{1/2}

as with normal products of powers :smile:



Unfortunately, that is what I thought which means something else has gone wrong, time for attempt number 3!
Reply 3
Avatar for davros
davros
16
Original post by BryonyG98
Unfortunately, that is what I thought which means something else has gone wrong, time for attempt number 3!


If you're still stuck, post your working.

I'll be offline soon but I'm sure someone else will be able to advise :smile:
Reply 4
Avatar for BryonyG98
BryonyG98
OP
3
What would x^1/2 * y^-1/2 equal... is isnt just -√(xy) ​is it?
Reply 5
Avatar for BryonyG98
BryonyG98
OP
3
Is it √y / ​√x?
Reply 6
Avatar for davros
davros
16
Original post by BryonyG98
What would x^1/2 * y^-1/2 equal... is isnt just -√(xy) ​is it?


x1/2y−1/2=xyx^{1/2}y^{-1/2} = \sqrt{\dfrac{x}{y}}

I'm going out shortly, but it might help if you posted your answer and the answer you're supposed to get.

Bear in mind that there may be several different equivalent answers because you can rationalize things or substitute from the original equation you're given to express xy\sqrt{xy} in terms of other things!
Reply 7
Avatar for BryonyG98
BryonyG98
OP
3
Original post by davros
x1/2y−1/2=xyx^{1/2}y^{-1/2} = \sqrt{\dfrac{x}{y}}

I'm going out shortly, but it might help if you posted your answer and the answer you're supposed to get.

Bear in mind that there may be several different equivalent answers because you can rationalize things or substitute from the original equation you're given to express xy\sqrt{xy} in terms of other things!


Thank you very much, I will have one more try and then post my answer and the answer from the book (:
Reply 8
Avatar for BryonyG98
BryonyG98
OP
3
Okay so for my answer I get:

-1-√y
√x
2y+√x
 √y

And the answer in the back of the book is:

2√(xy) + y
4y​√(xy) + x
Reply 9
Avatar for Laz236
Laz236
1
Replace √XY with X^(1/2)Y^(1/2) and differentiate giving you...
1/2 X^(-1/2) Y^(1/2) + 1/2 X^(1/2) Y^(-1/2) DY/DX + 1 + 2Y DY/DX = 0
Factories by the DY/DX and move any other terms other to the other side
DY/DX (1/2 X^(1/2) Y^(-1/2) + 2Y) = -1 - 1/2 X^(-1/2) Y^(1/2)
Divide by the terms on the left hand side to get DY/DX on its own and then simplify the equation to remove the fractions, although when im done i get the same answer in the book but with negatives on the top like...
-2√(xy) - y
4y​√(xy) + x
(edited 9 years ago)

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