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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