[scrabblehack]

I doubt it...it seems a whole lot more awkward.
I was trying to derive it by the derivative of the quotient but it doesn’t seem to work for me.

[johnnyb_61820]

So, to start with, do differentials rather than derivatives (you can probably do it with derivatives, but more steps). So, for instance, d(x^2) = 2x dx.

Now, the quotient rule is d(u/v) = (v du - u dv) / v^2. Therefore,

d(dy/dx) = (dx d(dy) - dy d(dx)) / dx^2

Now, the derivative is the differential divided by dx, so that puts another dx on the bottom

(d(dy/dx))/dx = (dx d(dy) - dy d(dx)) / dx^3

Now, just distribute the numerator and simplify:

(d(dy/dx))/dx = d(dy)/dx^2 - (dy/dx) (d(dx)/dx^2)

d(dy) = d^2y, so

(d(dy/dx))/dx = d^2y/dx^2 - (dy/dx) (d^2x/dx^2)

That’s the new formula.