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

1 Answer
Jan 2, 2016

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

Explanation:

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))