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. $x^\frac{2}{3} + y^\frac{2}{3} = 4$, $( -3\sqrt{3} ,1)$

Answer:

Equation of tangent is y =  $\frac{1}{\sqrt{3} }$x  + 4

Step-by-step explanation:

We are given the equation

$x^\frac{2}{3} + y^\frac{2}{3} = 4$,  

upon differentiating

d($x^\frac{2}{3} + y^\frac{2}{3} = 4$) /dx = d(4)/dx

$\frac{2}{3}x^-\frac{1}{3 } + \frac{2}{3}y^ -\frac{1}{3} dy/dx = 0$

dy/dx = $-x^\frac{1}{3} /y^-\frac{1}{3}$ = $-y^\frac{1 }{3}/ x^\frac{1}{3}$

upon substituting the values (x, y)= $( -3\sqrt{3} ,1)$

dy/dx = $\frac{1}{\sqrt{3} }$

equation of the tangent

y - 1 = $\frac{1}{\sqrt{3} }$ ( x- (-$3\sqrt{3}$))

y =  $\frac{1}{\sqrt{3} }$x  + 4