How do you specify whether you are evaluating some derivative at some points or you are differentiating a composition of functions? When one first learns about single variable differentiation, one is taught, for instance, about the distinction between $(f(x^2+1))'$ (differentiating the composition of the functions $f$ and $x^2+1$) and $f'(x^2+1)$ (evaluating the derivative of $f$ at $x^2+1$). There is obviously a distinction. I mean, if $f(x)=x$, the former qquals $2x$, while the latter equals $1$.
Now I would like to know how do you make this distinction in multivariable calculus. For instance, say we have the function $f(x, y)=x+y$ and we want to evaluate the derivative with respect to the first variable (which is $x$) at the points $(x, y)$ where $y=x^2$. I would just write $\frac{\partial f}{\partial x}(x, x^2)=1$. But couldn't $\frac{\partial f}{\partial x}(x, x^2)$ be also interpreted as the partial derivative of the function $x+x^2$ with respect to $x$ (which in fact is just an ordinary derivative, but equals $1+2x$, not $1$)? Is it assumed that $\frac{\partial f}{\partial x}(x, x^2)$ means the derivative with respect to $x$ evaluated at the points $(x, x^2)$? If this is the case, how would you denote things if you wanted to be in the second case, i.e. the one where you differentiate a composition of functions? This has only occured to me now even though I took multivariable calculus last semester.
 A: You are correct: $\frac{\partial f}{\partial x}(x, x^2)$ is ambiguous.
Depending on the context, there are different possible notations.  For example, if $f$ is a function of two variables, $f(x,y)$, I may write $f_1$ and $f_2$ for its two partial derivatives.  Thus
$$
\frac{\partial}{\partial x}\;f(x,x^2) =
f_1(x,x^2) + 2xf_2(x,x^2) .
$$
Another scheme seen sometimes: number the variables and write
$$
f^{(1,0)}(x_1,x_2) = \frac{\partial}{\partial x_1}f(x_1,x_2)
\quad\text{and}\quad
f^{(0,1)}(x_1,x_2) = \frac{\partial}{\partial x_2}f(x_1,x_2)
\\
\text{and in general,}\qquad f^{(i,j)}(x_1,x_2) = 
\frac{\partial^i}{\partial x_1^i}\frac{\partial^j}{\partial x_2^j}f(x_1,x_2)
$$
A: Good question! I will give you a quote from Analysis II by Terence Tao (this is on page 547; I don't know if this book has several editions, but please let me know in case there are and this reference is not accurate):

We sometimes replace the variables $x_j$ in $\frac{\partial f}{\partial x_j}$ with other symbols. For instance, if we are dealing with the function $f(x, y)=(x^2, y^2)$, then we might refer to $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ instead of $\frac{\partial f}{\partial x_1}$ and $\frac{\partial f}{\partial x_2}$. (In this case, $\frac{\partial f}{\partial x}(x, y)=(2x, 0)$ and $\frac{\partial f}{\partial y}(x, y)=(0, 2y))$. One should caution however that one should only relabel the variables if it is absolutely clear which symbol refers to the first variable, which symbol refers to the second variable, etc.; otherwise one may become unintentionally confused. For instance, in the above example, the expression $\frac{\partial f}{\partial x}(x, x)$ is just $(2x, 0)$, however one may mistakenly compute $$\frac{\partial f}{\partial x}(x, x)=\frac{\partial}{\partial x}(x^2, x^2)=(2x, 2x);$$ the problem here is that the symbol $x$ is being used for more than just the first variable of $f$. (On the other hand, it is true that $\frac{d}{dx}f(x, x)$ is equal to $(2x, 2x)$; thus the operation of total differentiation $\frac{d}{dx}$ is not the same as that of partial differentiation).

This quote and this Wikipedia section both address your issue. According to them, no, in your example $\frac{\partial f}{\partial x}(x, x^2)$ couldn't be interpreted as the derivative of the composition of the two functions. But, as Tao says, there is a good chance that someone might interpret it like that, even though it is not common to do so. This is why it would be much better to take Tao's advice and not use the same symbol for two different things (even though, as you said, some people keep doing this and you need to be able to comprehend what they write, so bear in mind what this "overloaded" notation means), a piece of advice that applies to all of mathematics, not just to how we denote partial derivatives.
As for your second question, if you want to take the derivative of the composition of the two functions, my humble opinion is that it's best to define a new function, say $\varphi(x)=f(x, x^2)=x+x^2$, and simply compute the derivative of this function $\varphi$. In this way there is no chance that someone is confused about what everything stands for.
