Question

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

Answer 1

`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))`