One of the problems arising in implicit differentiation is that we often get results that appear totally different depending on the approach we took
I would have cross-multiplied first to get
x^3y^2 + x^2y^2 = y+1
then
x^3(2y)dy/dx + 3x^2y^2 + x^2(2y)dy/dx + 2xy^2 = dy/dx
bring all dy/dx terms to the left, everybody else to the right, factor out dy/dx
dy/dx(2x^3 y +2x^2y - 1) = -3x^2y^2 - 2xy^2
dy/dx = (-3x^2y^2 - 2xy^2)/(2x^3 y + 2x^2 y - 1)
if you find dy/dx directly using the quotient rule
x^2(2y)(dy/dx) + 2xy^2 = ( (x+1)dy/dx - y - 1)/(x+1)^2
now it is harder to "get at" the dy/dx, so I multiplied both sides by (x+1)^2 , then factored out the dy/dx to get an answer of
dy/dx = ( -2xy^2(x+1)^2 - y -1)/( (x+1)(2x^2y - 1))
I admit that neither of my answers look like their answer, but they could also be correct
To test for the reasonableness of the answers being correct pick any point which lies on the original curve and evaluate the different dy/dx
It was easy to see that (1,1) was a point on the curve.
evaluating the different dy/dx , I found -5/3 for both of my answers, I also got -5/3 for their answer.
That does not "prove" that the answers are equivalent , (I would have to try "every" point), but it shows a high probability that they are correct.
BTW, (1, -1/2) is another point on the curve, you might want to check your answer against the above to see if you get the same dy/dx for that point
Differentiate x^2*y^2 = (y+1)/(x+1) in terms of x and y.
PS My answer is unfortunately at odds with that provided by the authors of the book (i.e. -y (y+1) (3x+2) all over x (x+1) (y+2), and I don't know whether it's a misprint or my method is wrong.
2 answers
I also get a much more complicated answer:
2xy^2 + 2x^2yy' = y'/(x+1) - (y+1)/(x+1)^2
y' = -(2x y^2 (x+1)^2 + (y + 1)) / (2x^2 y (x+1)^2 - (x+1))
2xy^2 + 2x^2yy' = y'/(x+1) - (y+1)/(x+1)^2
y' = -(2x y^2 (x+1)^2 + (y + 1)) / (2x^2 y (x+1)^2 - (x+1))