What is the implicit derivative of #5=xy^3-3xy#?

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Answer 1 · 2016-01-02T03:32:34

#f'(x) = dy/dx = (y(y^2-3))/(3x(1-y^2))#

Given: #5 = xy^3 -3xy#

We will find the derivative with respect to #dx#

#d/dx(5) = d/dx(xy^3) -d/dx(3xy) " " " " " # Product rule

#0 = d/dx(x) y^3 + x d/dx(y^3) - d/dx((3xy) - 3x(d/dx y) #

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

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

Gather like terms #dy/dx# on one side, then factor, solve for #dy/dx#

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

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

#dy/dx = (y(y^2-3))/(3x(1-y^2))#