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.
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
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))
To differentiate the equation x^2*y^2 = (y+1)/(x+1) with respect to both x and y, we can use the chain rule for differentiation.
Let's break down the process step by step:
1. Start by expressing the equation explicitly in terms of y.
x^2 * y^2 = (y + 1) / (x + 1)
2. Multiply both sides of the equation by (x + 1) to eliminate the denominator.
x^2 * y^2 * (x + 1) = y + 1
3. Expand the equation.
x^2 * y^2 * x + x^2 * y^2 = y + 1
4. Rearrange the terms so that y and its derivatives appear on one side of the equation.
x^3 * y^2 + x^2 * y^2 - y - 1 = 0
5. Now, differentiate both sides of the equation with respect to x while treating y as a function of x. Remember to apply the chain rule.
For the terms involving y^2:
d/dx(x^3 * y^2) = 3x^2 * y^2 + x^3 * 2y * dy/dx
For the terms involving y:
d/dx(x^2 * y^2) = 2x * y^2 + x^2 * 2y * dy/dx
For the constant term -1:
d/dx(-1) = 0
Thus, the derivative of the left-hand side is:
3x^2 * y^2 + x^3 * 2y * dy/dx + 2x * y^2 + x^2 * 2y * dy/dx
The derivative of the right-hand side, which is 0, will be 0 as well.
6. Simplify the equation by collecting like terms:
3x^2 * y^2 + x^2 * y^2 + 2x * y^2 = -2x^3 * y * dy/dx
7. Factor out the common terms on the left-hand side:
(3x^2 + x^2 + 2x) * y^2 = -2x^3 * y * dy/dx
8. Divide both sides by (x^2 * y) to solve for dy/dx:
-2x^3 * dy/dx = (3x^2 + x^2 + 2x) / (x^2 * y)
9. Multiply both sides by -1/2x^3 to obtain the final result:
dy/dx = -[(3x^2 + x^2 + 2x) / (2x^3 * y)]
Therefore, the derivative of x^2*y^2 = (y+1)/(x+1) in terms of x and y is dy/dx = -[(3x^2 + x^2 + 2x) / (2x^3 * y)].