This is a follow-up question to my previous question, Riemann integral interval confusion, from the same book.
Below is a snippet of context from the book:
Claim (I should probably be using $\frac{\mathrm{d}\phi_1}{\mathrm{d}a_i}$ here instead of $\frac{\mathrm{d}\phi_1}{\mathrm{d}x}$, but I'm not entirely sure what the correct notation would be):
$\frac{\mathrm{d} \phi_1}{\mathrm{d} x}\Big|_{a_i}=\lim_\limits{\Delta a\rightarrow 0}\left|\frac{\phi_1(a_{i+1})-\phi_1(a_{i})}{\Delta a}\right|$
These are my thoughts on this matter: For $\Delta a$ to even be permitted to approach $0$: $a_{i+1}$, $a_{i}$, or both must be changing as $\Delta a=a_{i+1}-a_{i}$ (I assume).
I think we can assume $a_i$ is fixed/constant, as we want to evaluate the derivative at $a_i$. This implies that $a_{i+1}$ must be changing (fact 1)
However, to evaluate $\frac{\mathrm{d} \phi_1}{\mathrm{d} x}\Big|_{a_{i+1}}$, we need $a_{i+1}$ to be fixed, which is a contradiction with (fact 1).
Hence, $\frac{\mathrm{d} \phi_1}{\mathrm{d} x}\Big|_{a_i}$ is ill-defined.
What is going on here? It seems that this might only make sense if lose the requirement that $a_j,\;i\neq j$ is fixed. But then we only know $\frac{\mathrm{d} \phi_1}{\mathrm{d} x}\Big|_{a_i}$, which is useless when computing the sum over all $a_i,\;\forall i$.
Is this just a non-rigorous leap, which is used to provide intuition as to why there is a "derivative" factor needed to "correct" the integral under parameterisation? Or... is there a flaw in my reasoning, and the derivative is well defined?