I'm no expert on Alan Turing at all, and the following will not directly answer your question, but it might give some context. Following the link provided, I also found the following page, which might shed some more light on things:

It says the following (my apologies for any errors in transcribing, I'm grateful for any corrections):

A formal expression $$ f(x) = \sum_{i=1}^n \alpha_i x^i $$ involving
the `indeterminate' (or variable) $x$, whose coefficients $\alpha_i$
are numbers in a field $K$, is called a ($K$-)polynomial of formal
degree $n$.

The idea of an `indeterminate' is distinctly subtle, I would almost
say too subtle. It is not (at any rate as van der Waerden [link added by me] sees it) the
same as variable. Polynomials in an indeterminate $x$, $f_1(x)$ and
$f_2(x)$, would not be considered identical if $f_1(x)=f_2(x)$ [for]
all $x$ in $K$, but the coefficients differed. They are in effect the
array of coefficients, with rules for multiplication and addition suggested
by their form

I am inclined to the view that this is too subtle and makes an inconvenient
definition. I prefer the indeterminate $x$ [?] be just the variable.

I think one thing to keep in mind here is that at this time anything relating to `computability' was not as clear as today. After all, Turing (an Church & Co.) were just discovering the essential notions.

In particular questions of intensionality vs. extensionality could have been an issue. It might be that Turing was pondering on the difference between functions (and also operations on functions) from a purely mathematical point of view (i.e., functions as extensional objects) vs. a computational point of view (i.e., functions as some form of formal description of a calculation process, which a priori can not be looked at in an extensional way).

All of this can be still seen in the context of the foundational crisis of mathematics (or at least strong echoes thereof). Related to this, are of course, questions of rigour, formalism, and denotation. This, in turn, is where your quote comes in. As others have outlined, Turing might have asked the question, what $\frac{\mathrm{d}y}{\mathrm{d}x}$ is (from a formal point of view), but also what is denoted by it, and (to his frustration) found that the answer to his question was not as clear as he wanted it to be.