Why the inconsistent derivatives?

November 3, 2007 1:01 AM

Where am I going wrong with this (very simple) implicit derivative?

I have an equation: xy + y^2/x = 2. I solve it this way:

(1 * y) + (x * dy/dx) + (2y*dy/dx * x^-1) + (y^2 * -x^-2) = 0
dy/dx(x + 2y/x) = y^2/x^2 - y
dy/dx((x^2 + 2y)/x) = (y^2 - x^2 * y)/x^2
dy/dx = (y^2 - x^2 * y)/(x^2) * x/(x^2 + 2y)
dy/dx = (y^2 - x^2 * y)/(x^3 + 2xy)

Now, the answers I have before me say the correct method is:

x^2 * y + y^2 = 2x
(2x * y) + (dy/dx * x^2) + (2y * dy/dx) = 2
dy/dx(x^2 + 2y) = 2 - 2xy
dy/dx = (2 - 2xy)/(x^2 + 2y)

Both of these appear correct to me, but the end result is different in each case? Can someone explain the ludicrously obvious mistake I am making?

posted by PuGZ to education (3 comments total)

No, there is no mistake. The answers are the same. In the second result, substitute xy + y^2/x in for the 2 in the numerator, then simplify. You will get the first result.
posted by number9dream at 1:36 AM on November 3, 2007

Your solution's fine, and preferable in one sense which I'll get to in a minute. Number9dream has the gist, which is a common thing in implicit differentiation - often, once the differentiation's complete (and you've isolated dy/dx), you may recognize some terms in the result that occured in the original defining equation of the curve as well - typically, one side of the original equation. That means you can, if it simplifies things, replace those terms with the equivalent expression from the other side of the equation. Like, if the original curve has the form y²=(an enormous expression), and (that enormous expression) occurs in the formula that results from differentiating, you can replace it with y² again.
More than you wanted to know:
Multiplying through to clear out denominators (the first step in the second solution) is somewhat dangerous, because it often changes the solution set of the equation, and you have to pay attention to that. In this example, it's not a big deal, but you can still see what I'm talking about: There is not any solution to the equation xy+y²/x=C with x=0 (for any constant C on the right), because the lefthand side is undefined when x=0. So dy/dx for the curve shouldn't be defined when x=0, and your solution reflects that, since x is a factor of the denominator. The second solution changes the domain for dy/dx.
posted by Wolfdog at 2:27 AM on November 3, 2007 [1 favorite]

Thank you very much to both of you. What a wonderful resource!

Wolfdog: Thank you for the extended response. It is something I should have remembered, but had slipped my mind. I imagine that there will indeed be such corner-cases in my exam this coming Monday and I very much appreciate the reminder: I'll be sure not to forget it!
posted by PuGZ at 3:40 AM on November 3, 2007