TITLE: Can a divisor of derivation be cancelled out
QUESTION [0 upvotes]: 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:
$$
\frac{\partial N}{\partial t} = \frac{\partial\phi}{\partial x}
$$

REPLY [1 votes]: 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.
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