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.