Question

If `(x-y)^3 = A(x + y)`, how do you prove that `(2x + y)dy/dx = x + 2y`?

Answer 1

If `(x-y)^3 = A(x + y)`, prove that `(2x + y)dy/dx = x + 2y`

Before differentiating, let's isolate the constant `A`

`(x-y)^3/(x+y) = A`

Now, use the quotient rule to differentiate implicitly:

`(3(x-y)^2(1-dy/dx)(x+y) - (x-y)^3(1+dy/dx))/(x+y)^2 = 0`.

Clearly we do not need the denominator, and to minimize notation,
let us use `D = (x-y)` and `S = (x+y)`, we need to solve :

`3D^2S(1- dy/dx) - D^3(1+dy/dx) = 0` for `dy/dx`.

Now divide all by `D^2`, to get:

`3S(1- dy/dx) - D(1+dy/dx) = 0`. So, we have:

`3S - 3S dy/dx - D -D dy/dx = 0`. And this tells us that:

`3S-D = (3S+D) dy/dx`

Return to `D = (x-y)` and `S = (x+y)`, and simplify (divide all by 2).