Question

How do you implicitly differentiate `-y=y-sqrt(x^3y-y)`?

Answer 1

`dy/dx =(3yx^2)/ ((4sqrt(x^3y-y))(x^3-1))`

Explanation:

implicit differentiation is derived from the following
`d/dx = d/dy *dy/dx`

so we need to only solve
`d/dy = 2- 1/(2sqrt(x^3y-y))(x^3-1) = (4sqrt(x^3y-y))/(2sqrt(x^3y-y))(x^3-1)`

`d/dx = - (3yx^2)/(2sqrt(x^3y-y))`

now lets divide and solve

`dy/dx = (2sqrt(x^3y-y))/((4sqrt(x^3y-y))(x^3-1)) *(3yx^2)/ (2sqrt(x^3y-y))`

`dy/dx =(3yx^2)/ ((4sqrt(x^3y-y))(x^3-1))`