What is the Implicit Derivative at a Given Point?

[callmelish]

Homework Statement

"Find dy/dx at the given point by using implicit differentiation"

x²y + y²x = -2 at (2, -1)

and

(x+y)³ = x³ + y³

Homework Equations

The Attempt at a Solution

1) x²(dy/dx) + y(2x) + y²(1) + 2y(dy/dx)(x) = -2
x²(dy/dx) + 2xy + y² + 2xy(dy/dx) = -2
dy/dx(x² + 2xy) = 2xy + y² -2
dy/dx = (2xy + y² -2)/(x² + 2xy)

dy/dx at (2, -1) = (2*2*-1 - 2)/(2² + 2*2*-1)
= 0/0 = 0

The first one has me confused, and since the second one is similar I didn't want to attempt it in case I'm completely wrong.

 

[Asphyxiated]

"x^2y + y^2x = -2 at (2, -1)"

When you take the derivative of this, whether implicit or not, -2 will fall out of the equation (go to zero).

 

[dextercioby]

Don't know that

$$\frac{dy}{dx} = -\frac{\frac{\partial F(x,y)}{\partial x}}{\frac{\partial F(x,y)}{\partial y}}$$

, where F(x,y)=0 is the implicit equation ?

 

[eumyang]

1) x²(dy/dx) + y(2x) + y²(1) + 2y(dy/dx)(x) = -2

The bolded should be 0 (the derivative of -2 is 0).

x²(dy/dx) + 2xy + y² + 2xy(dy/dx) = -2
dy/dx(x² + 2xy) = 2xy + y² -2

Both of the bolded terms here should be negative.

 

[osnarf]

What is the derivative of the constant function f(x) = -2? Don't forget you have to differentiate both sides of the equation.

 

[Asphyxiated]

Also it may help to look at an implicit plot of this function, at (2,-1) the slope is asymptotic.