Can a divisor of derivation be cancelled out

Question

Suppose I have this equation:

$$ \frac{\partial N}{\partial x \partial t} = \frac{\partial^2\phi}{\partial x^2} $$

Can I cancell $\partial x$ and become:

$$ \frac{\partial N}{\partial t} = \frac{\partial\phi}{\partial x} $$

Answer 1

No, you cannot. The closest you can get is to rewrite it as

$$ \frac{\partial }{\partial x }\left(\frac{\partial N}{\partial t} - \frac{\partial\phi}{\partial x}\right) = 0 $$ and "integrate it once" to get

$$\frac{\partial N}{\partial t} - \frac{\partial\phi}{\partial x} = C(t) $$ for some function $C(t)$. But this is really only roughly valid, and you'll need to have some additional assumptions/information about the differentiability of $N$ and $\phi$ and their domains in order to make it more rigorous.