Question

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

x^2/3 +y^2/3 =4

(-3sqrt(3),1)
(astroid)

Answer 1

If it is an astroid, then it should be

x^(2/3) +y^(2/3) =4
the way you typed it would result in a circle
http://www.wolframalpha.com/input/?i=x%5E2%2F3+%2By%5E2%2F3+%3D4

(2/3)x^(-1/3) + (2/3)y^(-1/3) dy/dx = 0
x^(-1/3) + y^(-1/3) dy/dx = 0
dy/dx = -(x^(-1/3))/y^(-1/3)
= -(x/y)^(-1/3)
= -(y/x)^(1/3)

at the point (-3√3 , 1)
dy/dx = -(2/(-3√3))^(1/3)

equation of tangent:
y-1 = -(2/(-3√3))^(1/3)(x + 3√3)
simplify if you have to