# How do you use implicit differentiation to find dy/dx given 2xy^2-x^3y=0?

Sep 13, 2016

See the explanation

#### Explanation:

Rearranging,

$\left(x\right) \left(y\right) \left(2 y - {x}^{2}\right) = 0$

This is a compounded equation for the three separate equations

x = 0, representing the y-axis and $y ' = \frac{1}{\frac{\mathrm{dx}}{\mathrm{dy}}} = \frac{1}{0} = \infty$

y = 0, representing the x-axis and y' = 0.

$y = {x}^{2} / 2$ representing a vertical parabola and y'=2x/2=x.

I think that the question could have been worded as follows.

How do you find y', given 2xy^2-x^3y=0?.

Of course, without regard to the stated aspects,

$2 {y}^{2} + 4 x y y ' = 3 {x}^{2} y + {x}^{3} y ' .$.

Separating y',

y'=(y(3x^2-2y))/(x(4y-x^2)

I request readers to compare both approaches to this problem.