Implicit equations: an example

Differential Calculus Grinshpan

Implicit equations: an example

The equation x2 - xy + y2 = 3 describes an ellipse.

dy Let's find out what the derivative is by implicit differentiation:

dx 2x - (y + xy ) + 2yy = 0

(-x + 2y)y = y - 2x

y - 2x 2x - y

y=

=

2y - x x - 2y

d2y We may as well find the second derivative .

dx2

Here is one way:

2x - y y=

x - 2y

(2 - y )(x - 2y) - (2x - y)(1 - 2y ) =

(x - 2y)2

xy - y = 3 (x - 2y)2

=

3

x

2x-y x-2y

-y

(x - 2y)2

2x2 - yx - xy + 2y2 =3

(x - 2y)3

x2 - xy + y2 = 6 (x - 2y)3

18 = (x - 2y)3

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