MATH SOLVE

3 months ago

Q:
# 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)

Accepted Solution

A:

The complete question is Use implicit differentiation to find an equation of the tangent line to the curve at the given point. [tex]x^\frac{2}{3} + y^\frac{2}{3} = 4[/tex], [tex]( -3\sqrt{3} ,1)[/tex]Answer:Equation of tangent is y = [tex]\frac{1}{\sqrt{3} }[/tex]x + 4 Step-by-step explanation:We are given the equation [tex]x^\frac{2}{3} + y^\frac{2}{3} = 4[/tex], upon differentiating d([tex]x^\frac{2}{3} + y^\frac{2}{3} = 4[/tex]) /dx = d(4)/dx [tex]\frac{2}{3}x^-\frac{1}{3 } + \frac{2}{3}y^ -\frac{1}{3} dy/dx = 0[/tex]dy/dx = [tex]-x^\frac{1}{3} /y^-\frac{1}{3}[/tex] = [tex]-y^\frac{1 }{3}/ x^\frac{1}{3}[/tex]upon substituting the values (x, y)= [tex]( -3\sqrt{3} ,1)[/tex]dy/dx = [tex]\frac{1}{\sqrt{3} }[/tex]equation of the tangent y - 1 = [tex]\frac{1}{\sqrt{3} }[/tex] ( x- (-[tex]3\sqrt{3}[/tex]))y = [tex]\frac{1}{\sqrt{3} }[/tex]x + 4