Question Involving implicit Differentiation

[skies222]

Hi!
I have a problem here that's been bugging me. I was wondering if anyone can give insight into where I'm going wrong

implicit differentiation problem

1) (x^2+y^2)/(x+y)=xy-2

find derivative (dy/dx) at point (-1, -1)


I know the basic premise. I used the quotient rule to find the derivative of the fraction. I am confused about the resulting fraction though. Once you find the derivative of the fraction, what should you do about the denominator ( which will be (x+y)^2 ). do you multiply each side by it? ( which would remove it from the left hand side, but add it to the right hand side). I've also tried moving the "xy" from the right hand side to the left side, but it seems to make the problem more confusing. Once I find the derivative (dy/dx) I know you just plug in the (x,y) value, so that's no problem.

If anyone can help, I would appreciate it greatly!

UPDATE!

I think I may have made the problem easier.. still going to try it out though. What if you got rid of the fraction on the left hand side from the beginning? Would that make it easier?

 

[neutrino]

Why not plug in the values for x and y and then solve for dy/dx?

 

[skies222]

Im not too sure what you mean? You mean plug in x= -1 and y=-1 into the equation? How would you take the derivative of that though? If you plug in numbers for the variables, you will come out with a number... a constant. The derivative of a constant is just zero... I am pretty sure (though I could be wrong) you plug in (-1,-1) at the end.

In case you missed my update, I think i know what i may have been doing wrong. I am going to try to get rid of the fraction from the beginning before i differntiate...may make it easier to differentiate.

 

[neutrino]

I meant that you plug them in AFTER you implicitly differentiate that expression. They you'll be left with a bunch of constants and the some dy/dx's.

 

[skies222]

Oh I see, I am sorry for the misunderstanding. Yea, that's what I am trying to do right now. Essentially, I am trying to get it to be (dy/dx)= (variables here)

I think I may have figured it out. After going through, I get -3/3 when I plug in (-1,-1) so, in other words I get -1 as my answer. Anybody try it and get this?

thanks for the reply too!