Find by implicit differentiation: x^2 + xy - y^2 = 4

Question

Find by implicit differentiation: x^2 + xy - y^2 = 4

Answer 1

Mia,

 

I am assuming you are looking for the derivative of y with respect to x - dy/dx.

 

Implicit differentiation allows us to find dy/dx even though we have not solved the equation for y. The differentiation is a bit more complex. We treat each y with the chain rule.

 

2x + x (dy/dx) + y -2y(dy/dx)=0

 

Note that x (dy/dx) + y comes from the xy term in the original equation. The right hand side is zero as the derivative of any constant with respect to x (or y) is 0.

 

We now just have algebra to isolate the dy/dx term.

 

 2x + x (dy/dx) + y -2y(dy/dx)=0

-2x                     -y                 = -2x - y

_________________________________

         x(dy/dx)         -2y(dy/dx) = -2x - y

 

factor out the dy/dx

 

         (dy/dx)(x-2y)=-2x-y

 

then divide through by x-2y

 

         dy/dx =(-2x-y)/(x-2y)

 

this result is correct if a but confusing. I would multiple the top and bottom of the right hand side by -1

 

         dy/dx = (2x+y)(2y-x)