Implicit Differentiation problem?

Question

Here's the problem:

Regard y as the independent variable and x as the dependent variable, and use the implicit differentiation to find dx/dy in terms of x and y.

(x^2 + y^2)^2 = 4(x^2)y

the correct answer was

(4x^2 - 4y (x^2 +y^2)) / (2x (2 (x^2 + y^2)-4y))

Answer 1

whoa some hard homework

Answer 2

2(x^2 + y^2)(2x + 2y dy/dx) = 8xy + 4x^2 dy/dx

Expand left side:

4(x^2+y^2)x + 4(x^2+y^2)y dy/dx = 8xy + 4x^2 dy/dx

Rearrange to put dy/dx on one side:

[4y(x^2+y^2) - 4x^2] dy/dx = 8xy - 4x(x^2+y^2)

Solve for dy/dx:

dy/dx = [8xy - 4x(x^2+y^2)] / [4y(x^2+y^2) - 4x^2]

NOTE: as pointed out in another question, it was dx/dy that was requested, not dy/dx, and I read this question too quickly. I show how to compute dx/dy with the other question. The approach is the same, and the above work could easily be modified to obtain dx/dy.

You get
2(x^2 + y^2)(2x dx/dy + 2y) = 8xy dx/dy + 4x^2

and you solve for dx/dy from there.

Answer 3

In the first member you must use chain rule and in the second the product rule: note x´ = dx/dy

2(x^2+ y^2).(2xx´ + 2y) = 4(2xx´)y +4(x^2)1
expanding
4(x^3)x´ + 4(x^2)y + 4(y^2)xx´+4(y^3) = 8xyx´+ 4(x^2)
put x´ in evidence

( 4x^3 + 4(y^2)x - 8xy)x´ = 4x^2 - 4(x^2)y - 4y^3

and then x´= (4x^2 - 4(x^2)y -4y^3)/(4x^3 + 4(y^2)x -8xy)

Answer 4

dx/dy = ( x^2 + y^2 ) ^3 / 3 = 4xy^3 / 3

you add one to the power and divide it by what the new power is. Hope this is helpful. ;-)

Answer 5

2(x² + y²)² = 4x²y

2(x² + y²)(2x*(dx/dy) + 2y) = 8xy*(dx/dy) + 4x²
4(x² + y²)(x*(dx/dy) + y) = 4(2xy*(dx/dy) + x²)
(x³*(dx/dy) + x²y + xy²*(dx/dy) + y³) = (2xy*(dx/dy) + x²)
x³*(dx/dy) + xy²*(dx/dy) - 2xy*(dx/dy) = x² - x²y - y³
(dx/dy)*(x³ + xy² - 2xy) = x² - x²y - y³
dx/dy = (x² - x²y - y³) / (x³ + xy² - 2xy)