Recently the question If 𝑑𝑑⁢𝑥 is an operator, on what does it operate? was asked on mathoverflow.  It seems that some users there objected to the question, apparently interpreting it as an elementary inquiry about what kind of thing is a differential operator, and on this interpretation, I would agree that the question would not be right for mathoverflow. And so the question was closed down (and then reopened, and then closed again….sigh). (Update 12/6/12: it was opened again,and so I’ve now posted my answer over there.)

Meanwhile, I find the question to be more interesting than that, and I believe that the OP intends the question in the way I am interpreting it, namely, as a logic question, a question about the nature of mathematical reference, about the connection between our mathematical symbols and the abstract mathematical objects to which we take them to refer.  And specifically, about the curious form of variable binding that expressions involving 𝑑⁢𝑥 seem to involve.  So let me write here the answer that I had intended to post on mathoverflow:

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To my way of thinking, this is a serious question, and I am not really satisfied by the other answers and comments, which seem to answer a different question than the one that I find interesting here.

The problem is this. We want to regard 𝑑𝑑⁢𝑥 as an operator in the abstract senses mentioned by several of the other comments and answers. In the most elementary situation, it operates on a functions of a single real variable, returning another such function, the derivative. And the same for 𝑑𝑑⁢𝑡.

The problem is that, described this way, the operators 𝑑𝑑⁢𝑥 and 𝑑𝑑⁢𝑡 seem to be the same operator, namely, the operator that takes a function to its derivative, but nevertheless we cannot seem freely to substitute these symbols for
one another in formal expressions. For example, if an instructor were to write 𝑑𝑑⁢𝑡⁢𝑥3 =3⁢𝑥2, a student might object, “don’t you mean 𝑑𝑑⁢𝑥?” and the instructor would likely reply, “Oh, yes, excuse me, I meant 𝑑𝑑⁢𝑥⁢𝑥3 =3⁢𝑥2. The other expression would have a different meaning.”

But if they are the same operator, why don’t the two expressions have the same meaning? Why can’t we freely substitute different names for this operator and get the same result? What is going on with the logic of reference here?

The situation is that the operator 𝑑𝑑⁢𝑥 seems to make sense only when applied to functions whose independent variable is described by the symbol “x”. But this collides with the idea that what the function is at bottom has nothing to do with the way we represent it, with the particular symbols that we might use to express which function is meant.  That is, the function is the abstract object (whether interpreted in set theory or category theory or whatever foundational theory), and is not connected in any intimate way with the symbol “𝑥”.  Surely the functions 𝑥 ↦𝑥3 and 𝑡 ↦𝑡3, with the same domain and codomain, are simply different ways of  describing exactly the same function. So why can’t we seem to substitute them for one another in the formal expressions?

The answer is that the syntactic use of 𝑑𝑑⁢𝑥 in a formal expression involves a kind of binding of the variable 𝑥.

Consider the issue of collision of bound variables in first order logic: if 𝜑⁡(𝑥) is  the assertion that 𝑥 is not maximal with respect to <, expressed by ∃𝑦 𝑥 <𝑦, then 𝜑⁡(𝑦), the assertion that 𝑦 is not maximal, is not correctly described as the assertion ∃𝑦 𝑦 <𝑦, which is what would be obtained by simply replacing the occurrence of 𝑥 in 𝜑⁡(𝑥) with the symbol 𝑦. For the intended meaning, we cannot simply syntactically replace the occurrence of 𝑥 with the symbol 𝑦, if that occurrence of 𝑥 falls under the scope of a quantifier.

Similarly, although the functions 𝑥 ↦𝑥3 and 𝑡 ↦𝑡3 are equal as functions of a real variable, we cannot simply syntactically substitute the expression 𝑥3 for 𝑡3 in 𝑑𝑑⁢𝑡⁢𝑡3 to get 𝑑𝑑⁢𝑡⁢𝑥3. One might even take the latter as a kind of ill-formed expression, without further explanation of how 𝑥3 is to be taken as a function of 𝑡.

So the expression 𝑑𝑑⁢𝑥 causes a binding of the variable 𝑥, much like a quantifier might, and this prevents free substitution in just the way that collision does. But the case here is not quite the same as the way 𝑥 is a bound variable in ∫10𝑥3 𝑑⁢𝑥, since 𝑥 remains free in 𝑑𝑑⁢𝑥⁢𝑥3, but we would say that ∫10𝑥3 𝑑⁢𝑥 has the same meaning as ∫10𝑦3 𝑑⁢𝑦.

Of course, the issue evaporates if one uses a notation, such as the 𝜆-calculus, which insists that one be completely explicit about which syntactic variables are to be regarded as the independent variables of a functional term, as in 𝜆⁢𝑥.𝑥3, which means the function of the variable 𝑥 with value 𝑥3.  And this is how I take several of the other answers to the question, namely, that the use of the operator 𝑑𝑑⁢𝑥 indicates that one has previously indicated which of the arguments of the given function is to be regarded as 𝑥, and it is with respect to this argument that one is differentiating.  In practice, this is almost always clear without much remark.  For example, our use of 𝜕𝜕𝑥 and 𝜕𝜕𝑦 seems to manage very well in complex situations, sometimes with dozens of variables running around, without adopting the onerous formalism of the 𝜆-calculus, even if that formalism is what these solutions are essentially really about.

Meanwhile, it is easy to make examples where one must be very specific about which variables are the independent variable and which are not, as Todd mentions in his comment to David’s answer. For example, cases like

𝑑𝑑⁢𝑥⁢∫𝑥0(𝑡2+𝑥3)𝑑𝑡𝑑𝑑⁢𝑡⁢∫𝑥𝑡(𝑡2+𝑥3)𝑑𝑡

are surely clarified for students by a discussion of the usage of variables in formal expressions and more specifically the issue of bound and free variables.