Find the slope of the tangent line to the curve

sqrt(1+3x^2y^2) - 2xy = -1 at the point (1,4).

sqrt(1 + 3x^2 * y^2) = 2xy - 1
1 + 3x^2 * y^2 = 4x^2 * y^2 - 4xy + 1

Derive implicitly:

0 + 3 * (x^2 * 2y * dy + 2x * y^2 * dx) = 4 * (x^2 * 2y * dy + 2x * y^2 * dx) - 4 * (x * dy + y * dx) + 0
3 * (2x^2 * y * dy + 2xy^2 * dx) = 4 * (2x^2 * y * dy + 2xy^2 * dx) - 4 * (x * dy + y * dx)
3 * 2 * x * y * (x * dy + y * dx) = 4 * 2 * x * y * (x * dy + y * dx) - 4 * (x * dy + y * dx)
6 * xy * (x * dy + y * dx) = (8 * xy - 4) * (x * dy + y * dx)
0 = (8 * xy - 4 - 6xy) * (x * dy + y * dx)
0 = (2xy - 4) * (x * dy + y * dx)
0 = (xy - 2) * (x * dy + y * dx)

x = 1
y = 4

0 = (1 * 4 - 2) * (1 * dy + 4 * dx)
0 = (2) * (dy + 4dx)
0 = 2dy + 8dx
0 = dy + 4dx
dy = -4dx
dy/dx = -4

use implicit differentiation to find dy/dx at (1,4)
1/(2*sqrt(1+3x^2y^2))*(3*(2x*y^2+2x^2*… - 2*(x*dy/dx+y) = 0
1/(2*sqrt(49))*(3*(32+8*dy/dx)) - 2*(dy/dx+4) = 0
1/14*(96+24*dy/dx) - 2dy/dx - 8 = 0
12/7*dy/dx - 2dy/dx = 8 - 48/7
-2/7*dy/dx=8/7
dy/dx = -4