Find the derivative of the implicit function (X^2+Y^2)^3=8X^2Y^2.

Answer 1

To find the derivative of the implicit function, we can first rewrite it in terms of Y explicitly.

(X^2 + Y^2)^3 = 8X^2Y^2

Expand the left side:

(X^6 + 3X^4Y^2 + 3X^2Y^4 + Y^6) = 8X^2Y^2

Now, differentiate both sides of the equation with respect to X:

6X^5 + 12X^3Y^2 + 6XY^4 + 6Y^5 * (dY/dX) = 16XY^2 + 16X^2Y * (dY/dX)

Move all terms involving (dY/dX) to one side:

6X^5 + 12X^3Y^2 + 6XY^4 - 16XY^2 - 16X^2Y = -6Y^5 * (dY/dX) + 6Y^2 * (dY/dX)

Now, factor out the common factor of (dY/dX):

6X^5 + 12X^3Y^2 + 6XY^4 - 16XY^2 - 16X^2Y = (6Y^2 - 6Y^5) * (dY/dX)

Finally, divide both sides by (6Y^2 - 6Y^5) to solve for (dY/dX):

(dY/dX) = (6X^5 + 12X^3Y^2 + 6XY^4 - 16XY^2 - 16X^2Y) / (6Y^2 - 6Y^5)