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

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Answer 1 · 2016-09-13T16:24:16

See the explanation

Rearranging,

$(x)(y)(2y-x^2)=0$

This is a compounded equation for the three separate equations

x = 0, representing the y-axis and $y'=1/(dx/dy)=1/0=oo$

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,

$2y^2+4xyy'=3x^2y+x^3y'.$ .

Separating y',

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

I request readers to compare both approaches to this problem.

How do you use implicit differentiation to find dy/dx given 2xy^2-x^3y=02xy^2-x^3y=0?