$1)$ How do I denote derivative of $ax^2+b$ in terms of $ax^2$?
$(ax^2+b)'(ax^2)$ can easily be confused with $ax^2\cdot(ax^2+b)'$.
$2)$ How do I denote the derivative of $ax^2+b$ in terms of $ax^2$ at point $c$?
$3)$ How do I denote the derivative of $ax^2+b$ in terms of $x$ at point $c$?
$3)$ How do I denote the derivative of $f(g(x))$ in terms of $g(x)$ at point $a$?
I want to denote all of this without using $\text{d}$.
Ah. Where as the prime notation on a function symbol is taken as being with respect to the function's argument, the prime notation over an expression is taken as being with respect to the independent variable of the discussion (most usually that is either $x$ or $t$).
That is, if $f(x)=ax+b$, then $\;f'(ax^2) = \frac{\mathrm d f(ax^2)}{\mathrm d (ax^2)} = \frac{\mathrm d (ax^2+b)}{\mathrm d (ax^2)} \\[2ex] f(ax^2)' = (ax^2+b)' = \frac{\mathrm d (ax^2+b)}{\mathrm d x}$
So, you want to use the prime notation on an expression to indicate you are deriving with respect to another expression rather than the implicit variable itself.
$$[u\mapsto u+b]' (ax^2) = \left.\frac{\mathrm d u+b}{\mathrm d u}\right\vert_{u:=ax^2} = \frac{\mathrm d(ax^2+b)}{\mathrm d (ax^2)}$$
You could establish in your forward that you were using a subscripted dash notation.
$$(ax^2+b)'_{(ax^2)} = \frac{\mathrm d (ax^2+b)}{\mathrm d (ax^2)}$$
Or simply rely on the chain rule. $\frac{\mathrm d (ax^2+b)}{\mathrm d x}\frac{\mathrm d x}{\mathrm d (ax^2)} = \frac{(ax^2+b)'}{(ax^2)'}$