Vrftjkry

Is it possible to do some kind of simplifications on an expression like

$$
f : x, y to mathbb{R} \
frac{frac{partial^2 f}{partial y^2}}{frac{partial f}{partial y}} = frac{partial f}{partial y}
$$

So that $frac{partial f}{partial y}$ in the denominator reduces $frac{partial^2 f}{partial y^2}$ to $frac{partial f}{partial y}$ like $frac{a^2}{a}=a$, without knowing the function $f$.

I would say no, that's not allowed, but I'm just wondering.

Thank you.

$$frac{frac{partial^2 f}{partial y^2}}{frac{partial f}{partial y}}=frac{frac{partial}{partial y}left(frac{partial f}{partial y}right)}{frac{partial f}{partial y}}nefrac{partial f}{partial y}$$

just as

$$ frac{y^{primeprime}}{y^prime}ne y^prime $$

That is indeed not allowed, even for single-variable derivatives. Take for example $f(x) = x^2$. Then
$$ frac{f''}{f'} = frac{2}{2x} = frac{1}{x} not= 2x = f' $$

No. Remember the second derivative is just the derivative of the first derivative, so using $g$ for the first derivative, your equation is equivalent to $frac{partial g}{partial y}=g^2$, which of course can't always be true.

No, usually these kinds of analogies don't work. Take for example $f(x,y)=y$.