How do you implicitly differentiate #-1=y^3x-xy+x^4y #?

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Answer 1 · 2016-03-23T08:33:04

#dy/dx=(y-y^3-4yx^3)/(3xy^2-x+x^4)#

The equation is:

#y^3x-xy+x^4y=-1#

Differentiating w.r.t. #x#

We have to apply product rule:

#[y^3*d/dxx+xd/dx*y^3]-[x*d/dxy+yd/dx*x]+[x^4*d/dx*y+y*d/dxx^4]=0#

#[y^3+3xy^2dy/dx]-[x*dy/dx+y]+[x^4*dy/dx+4yx^3]=0#

#y^3+3xy^2dy/dx-xdy/dx-y+x^4dy/dx+4yx^3=0#

#3xy^2dy/dx-xdy/dx+x^4dy/dx=-y^3+y-4yx^3#

#dy/dx[3xy^2-x+x^4]=y-y^3-4yx^3#

#dy/dx=(y-y^3-4yx^3)/(3xy^2-x+x^4)#