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 (−3√3 , 1) (astroid)

Answer 1

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

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

y' = - x^(-1/3) /y^(-1/3) = - y^(1/3) / x^(1/3)

Substituting at the point (-3√3,1):

−(1)1/3(−3√3)1/3=−1(−1·3·31/2)1/3=−1(−1)1/3·(3)1/3·(31/2)1/3=−1−1·(3)1/3·31/6=131/2=1√3

Using point-slope equation: y−1 =1√3(x−(−3√3))

y=1√3x+1√3(3√3)+ 1

y=1√3x+ 4