Cancellation rules for partial derivatives

Question

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.

Answer 1

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

just as

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

Answer 2

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.

Answer 3

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

Answer 4

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