dy/dx(y=x/(x+y)) is y'=y/((x+y)^2+x)
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If I multiply both sides of the original equation by (x+y) to clear the fraction I get yx+y^2=x
dy/dx(yx+y^2=x_ is y'=(1-y)/(x+2y)
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Is y'=y/((x+y)^2+x) the same as y'=(1-y)/(x+2y)? I am having trouble showing that it is or is not?
Or am I doing something wrong by multiplying both sides by x+y?
Does your textbook actually use the notation "dy/dx(y=x/(x+y))"? It seems pretty horrible to me, but notations do vary around the world.
The important thing here is that
implicit differentiation is very sensitive to details. Generally, if you do anything to simplify the work, you will get a different form of answer, which is equivalent but can be very hard to confirm.
If you take your derivative and replace y with x/(x+y), then simplify, the result looks more like theirs. If you then set the two denominators you get equal to one another and manipulate to find the conditions under which they are equal, you will find that the definition of the function is just what you need. So I am convinced that they are equivalent.
You could also actually solve for an explicit function (which turns out to be a double-valued linear "function", if I did my work right), and then plug that into each form of the derivative to see if they are the same. I didn't try doing all that work.
Or, as a "spot check", you could find a specific (x,y) pair that satisfies the given relation, and put them into each derivative to see if you get the same number. You'll find something interesting if you do that, too.
Bottom line: if you are given such a problem on a test, do exactly what they say to do, without doing anything special, no matter how wise it may be.
