Determine the slope of the graph of 3(x^2+y^2)^2 = 100xy.

To determine the slope of the graph of the equation 3(x^2+y^2)^2 = 100xy, we first need to rewrite the equation in a form that allows us to find the slope.

Start by expanding the left side of the equation:

3(x^2+y^2)^2 = 3(x^4 + 2x^2y^2 + y^4)
= 3x^4 + 6x^2y^2 + 3y^4

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

d/dx [3x^4 + 6x^2y^2 + 3y^4] = d/dx [100xy]
12x^3 + 12xy^2 = 100y + 100x(dy/dx)
12x^3 + 12xy^2 = 100y + 100x(dy/dx)

Now, solve for dy/dx to find the slope:

dy/dx = (12x^3 + 12xy^2 - 100y) / 100x

Therefore, the slope of the graph of the equation 3(x^2+y^2)^2 = 100xy is given by (12x^3 + 12xy^2 - 100y) / 100x.

Given: 3(x^2 y^2)^2 = 100xy

Differentiating both sides with respect to x, we get:

6(x^2 y^2)^2 - (d/dx)(x^2 y^2) = 100 - (d/dx)(xy)

=> 6(x^2 y^2)^2 - (2xy^2 + 2x^2y) = 100(x - dy/dx + y)

=> 12x^3 y^4 + 12x^5 y^2 = 100x - dy/dx + 100y

=> 100x - dy/dx = 100y - 12x^3 y^4 - 12x^5 y^2

=> dy/dx = (y - (12x^3 y^4 + 12x^5 y^2)/100x)

The slope of the graph at the point (x, y) is given by:

m = dy/dx = (y - (12x^3 y^4 + 12x^5 y^2)/100x) at (x, y)