Shouldn't the function $g(x,y(x))=x^³ + y(x)$ satisfy $\frac{dy}{dx}=-\frac{g_x}{g_y}$? I get $g_x=3x^2 + \frac{dy}{dx}$ and $g_y=-1$ which does not satisfy it... Where do you think the mistake is?
Furthermore, is it possible to derive a general (so not for the particular function above) formula for $\frac{d^2y}{dx^2}$ in terms of partial derivatives of $g(x,y(x))$?
I can't get anything satisfying really. Thank you.
On have to differentiate the function $g(x,y)$ thanks to partial dérivatives :
$g(x,y(x))=x^3 + y(x)$
$\frac{∂g}{∂x}dx+\frac{∂g}{∂y}dy=3x^2dx+dy$
$(\frac{∂g}{∂y}-1)dy=(3x^2-\frac{∂g}{∂x})dx$
$\frac{dy}{dx} = \frac{3x^2-\frac{∂g}{∂x}}{\frac{∂g}{∂y}-1}$