Find the equation of the tangent line to the curve x^(3)+3xy+y^(3)=1 at the point (1,0).

Answer 1

Implicit derivative:

3x^2 + 3x dy/dx + 3y + 3y^2 dy/dx = 0
dy/dx(3x + 3y^2) = -3x^2 - 3y
dy/dx = (-x^2 - y)/(x + y^2)

at (1,0)
dy/dx = -1 - 0)/(1 + 0) = -1

let y = -x + b
at (1,0)
0 = -1 + b
b = 1

tangent equation at (1,0) is
y = -x + 1