Is the $x$ in $\frac{\mathrm{d}}{\mathrm{d}x}$ a symbol in the sense of Harper's PFPL?
Some mathematicians find it natural to substitute for the variable in the denominator of a derivative, writing things like $\frac{d \log V}{d\log p}$. This suggests that the $x$ in the denominator $\frac{dy}{dx}$ is neither bound nor binding nor a symbol. 

Rather $\frac{dy}{dx}$ seems to be an operation on "variable quantities" $y,x$ requiring some side conditions, similar to how the usual division $a/b$ requires $b\neq 0$, or how the operation $\frac{v}{v_1^0\ldots v_i^1\ldots v_n^0}$ in my comment requires $v_1,\ldots,v_n$ to be linearly independent and $v$ to lie in their span. 

I suspect that the side condition for $\frac{dy}{dx}$ should be analogous: $dx$ needs to be linearly independent and $dy$ has to be a multiple of $dx$. 

With these side conditions we are not allowed to substitute a constant for $x$ in $\frac{dy}{dx}$ since then $dx=0$ and so $dx$ is not linearly independent. That would explain why we are not allowed to write $\frac{d x^2}{d3}$ or to take the derivative of equation $x=0$ w.r.t. to $x$ to conclude $1=0$.

I'd be interested in hearing if someone sees an immediate problem with this interpretation. 

(I see some subtlety in the fact that $d(x|_{x=3})$ cannot mean the same as $(dx)|_{x=3}$, since otherwise I'd expect $\frac{dy}{dx}|_{x=3}$ to be the same as $\frac{d(y|_{x=3})}{d(x|_{x=3})}$ which would not be allowed by the side condition. On the other hand, that is not so strange from a differential geometric point of view if we read $|_{x=a}$ as restriction: there is a difference between restricting a differential form and pulling it back. But I need to think more about this.)