2.6.5 Implicit differentiation

[karush]

$\tiny{166.2.6.5}$
Find y'
$$x^2-4xy+y^2=4$$
dy/dx
$$2x-4(y+xy')+2yy'=2x-4y-4xy'+2yy'=0$$
factor
$$y'(-4x+2y)=-2x+4y=$$
isolate
$$y'=\dfrac{-2x+4y}{-4x+2y}
=\dfrac{-x+2y}{-2x+y}$$

typo maybe not sure if sure if factoring out 4 helped

 

[Prove It]

$\tiny{166.2.6.5}$
Find y'
$$x^2-4xy+y^2=4$$
dy/dx
$$2x-4(y+xy')+2yy'=2x-4y-4xy'+2yy'=0$$
factor
$$y'(-4x+2y)=-2x+4y=$$
isolate
$$y'=\dfrac{-2x+4y}{-4x+2y}
=\dfrac{-x+2y}{-2x+y}$

typo maybe not sure if sure if factoring out 4 helped

What you've done is correct.