If y = y(x), then the slope of the tangent line to (1, 1) is equal to the value of the derivative dy/dx when x = 1 and y = 1.
Compute the derivative using implicit differentiation:
d/dx [xy ^2 + y] = d/dx [2x]
d/dx [xy ^2] + d/dx [y] = 2 d/dx [x]
(x d/dx [y ^2] + d/dx [x] y ^2) + dy/dx = 2
2xy dy/dx + y ^2 + dy/dx = 2
(2xy + 1) dy/dx = 2 - y ^2
dy/dx = (2 - y ^2) / (2xy + 1)
Plug in x = 1 and y = 1 :
slope = dy/dx = (2 - 1^2) / (2*1*1 + 1) = 1/3
Now use the point-slope formula to get the equation of the line:
y - 1 = 1/3 (x - 1)
y = x/3 + 2/3