What is the partial derivative between two independent variables?

Question

If I have $f(x_1) = x_1^2$, what is $\frac{\partial f}{\partial x_2}$, given no other information? Is it 0?

Answer 1

If you want to be really proper, then it is undefined. Here, you seem to have a function $f:\Bbb{R}\to\Bbb{R}$, defined as $f(t)=t^2$ (or $f(x_1)=x_1^2$ as you write it) so you can only talk about its derivative $f'$ (or $\frac{df}{dt}$ or $\frac{df}{dx_1}$). But $\frac{\partial f}{\partial x_2}$ (which really ought to mean the partial derivative of $f$ with respect to the second entry) is just not defined.

However, what you can ask is to consider the different function, $F:\Bbb{R}^2\to\Bbb{R}$, defined as $F(x_1,x_2)=x_1^2$. Then, you can say $\frac{\partial F}{\partial x_2}=0$.

Though be warned that sometimes people abuse notation and say $f$ and $F$ are the same thing (even though they're really not), and by abuse of language and notation, write $\frac{\partial f}{\partial x_2}=0$.