Calculus: Early Transcendentals, Chapter 3, 3.5, Section 3.5, Problem 27

Note:- 1) If y = x^n ; then dy/dx = n*x^(n-1) ; where n = real number
2) If y = u*v ; where both u & v are functions of 'x' , then
dy/dx = v*(du/dx) + u*(dv/dx)
3) If y = k ; where 'k' = constant ; then dy/dx = 0
Now, the given function is :-
(x^2) + xy + (y^2) = 0
or, 2x + y + x*(dy/dx) + 2y*(dy/dx) = 0
Thus, putting x = y = 1 ; we get
3 + 3*(dy/dx) = 0
or,
or, dy/dx = -1 = slope of the tangent to the curve at (1,1)
Thus, equation of the tangent line to the given curve at the point (1,1) is :-
y - 1 = (-1)*(x - 1)
or, y -1 = -x + 1
or, x + y = 2 equation of the tangent