Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.
x^2y + y^2x = 3
d[x^2y + y^2x] = 0 -->
2xydx + x^2dy + 2xydy + y^2dx = 0 ---->
(2xy + y^2)dx + (2xy + x^2) dy = 0 ---->
dy/dx = -(2xy + y^2)/(2xy + x^2)
use the product rule for each term on the left side
x^2(dy/dx) + y(2x)(dx/dx) + y^2(dx/dx) + x(2y)(dy/dx) = 0
x^2(dy/dx) + 2xy + y^2 + 2xy(dy/dx) = 0
dy/dx is a common factor, so ...
dy/dx(x^2 + 2xy) = -2xy - y^2
dy/dx = (-2xy - y^2)/(x^2 + 2xy)
To find dy/dx using implicit differentiation, we'll differentiate each term with respect to x while treating y as a function of x. Let's go step by step:
1. Differentiate both sides of the equation with respect to x.
d/dx(x^2y + y^2x) = d/dx(3)
2. Apply the product rule to each term.
(d/dx(x^2y) + d/dx(y^2x)) = 0
3. Simplify the derivatives using the chain rule.
2xy + x^2(dy/dx) + 2yx + y^2 = 0
4. Combine like terms.
2yx + 2xy + x^2(dy/dx) + y^2 = 0
5. Rearrange the equation to solve for dy/dx.
2xy + 2yx + x^2(dy/dx) = -y^2
6. Subtract 2xy and 2yx from both sides.
x^2(dy/dx) = -y^2 - 2xy - 2yx
7. Simplify.
x^2(dy/dx) = -y^2 - 4xy
8. Finally, divide both sides by x^2 to solve for dy/dx.
dy/dx = (-y^2 - 4xy) / x^2
So, the derivative dy/dx is given by (-y^2 - 4xy) / x^2.
To find dy/dx using implicit differentiation, you'll need to differentiate both sides of the equation with respect to x, treating y as a function of x.
Let's start by differentiating the left side of the equation, which involves using the product rule and the chain rule.
Differentiating x^2y with respect to x:
(d/dx) (x^2y) = 2xy + x^2 * (dy/dx)
Differentiating y^2x with respect to x:
(d/dx) (y^2x) = y^2 * (dx/dx) + x * (d/dx) (y^2)
= y^2 + 2xy * (dy/dx)
Now, equating this with the derivative of the constant term on the right side of the equation (which is zero since the derivative of a constant is zero), we have:
2xy + x^2 * (dy/dx) + y^2 + 2xy * (dy/dx) = 0
Combining like terms, we get:
(2xy + 2xy) * (dy/dx) + x^2 * (dy/dx) + y^2 = 0
4xy * (dy/dx) + x^2 * (dy/dx) + y^2 = 0
Now, we can factor out (dy/dx):
(4xy + x^2) * (dy/dx) = -y^2
Finally, we can solve for (dy/dx):
dy/dx = -y^2 / (4xy + x^2)
So, the derivative dy/dx of the equation x^2y + y^2x = 3 is -y^2 / (4xy + x^2).