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

Question

x^2/3 + y^2/3 = 4, (-3sqrt3, 1)

Answer 1

You can actually find y' fairly easily explicitly.

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

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

y'=(3/2)(4-x^3/2)^1/2

But using implicit differentiation:

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

2/3y^1/3y'=-2/3x^1/3

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

y'(-3√3,1) = -(1/-3√3)^1/3≈0.57735

So the equation of the tangent line using point-slope:

y-1 = 0.57735(x+3√3)

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