Question

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2⁄3 + y2⁄3 = 4, −3 3 , 1 (astroid)

Answer 1

The complete question is

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)`

Answer:

Equation of tangent is y =
`(1)/(√(3) )`x + 4

Explanation:

We are given the equation

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

upon differentiating

d(
`x^(2)/(3) + y^(2)/(3) = 4`) /dx = d(4)/dx

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

dy/dx =
`-x^(1)/(3) /y^-(1)/(3)` =
`-y^(1 )/(3)/ x^(1)/(3)`

upon substituting the values (x, y)=
`( -3√(3) ,1)`

dy/dx =
`(1)/(√(3) )`

equation of the tangent

y - 1 =
`(1)/(√(3) )` ( x- (-
`3√(3)`))

y =
`(1)/(√(3) )`x + 4