Use implicit differentiation to find an equation of the tangent line to the curve x^2 + 2xy - y^2 + 7x = 8 at?

Question

Use implicit differentiation to find an equation of the tangent line to the curve x^2 + 2xy - y^2 + 7x = 8 at the point (1,2)
Give your answer in y = mx + b

Answer 1

derivating both sides
2x+2(y+xy´)-2yy´+7=0
at (1,2)

2+4+2y´-4y´+7=0
2y´=13 so y´=13/2 and the equation is

y-2=13/2(x-1) or y= 13/2x-9/2

Answer 2

x^2 + 2xy - y^2 + x = 6 2x * dx + 2x * dy + 2y * dx - 2y * dy + dx = 0 x = 2 y = 4 2 * 2 * dx + 2 * 2 * dy + 2 * 4 * dx - 2 * 4 * dy + dx = 0 4dx + 4dy + 8dx - 8dy + dx = 0 dx * (4 + 8 + a million) + dy * (4 - 8) = 0 13 * dx - 4 * dy = 0 13 * dx = 4 * dy 13/4 = dy/dx we make sure a line with a slope of 13/4 that passes by (2,4) 4 = (13/4) * 2 + b 4 = (13/2) + b 8/2 - 13/2 = b -5/2 = b y = (13/4) * x - (5/2)

Answer 3

2x+2y+2xy'-2yy'+7=0
For x=1 and y=2
2+4+2y'-4y'+7=0
y'=13/2
y=(13/2)x+b
Passes from (1,2)
2=13/2+b
b=-9/2
Finally
y=(13/2)x-9/2

Answer 4

y=4.5x-2.5