I'm a calculus noob learning over the internet. I think the best way to ask my question is just to put up a little diagram I made in paint.
Now this is my attempt to write the chain rule using d/dx notation:
So my question is can I use d/dx notation and signify that I'm passing g(x) into the derivative of f(x) without ambiguity as to what I'm doing?
To write Chain Rule using Leibniz's notation, we have: $$ \frac{\mathrm d}{\mathrm dx}[f(g(x))] = \frac{\mathrm d}{\mathrm dg(x)}[f(g(x))] \cdot \frac{\mathrm d}{\mathrm dx}[g(x)] $$ By letting $u = g(x)$ and $y = f(u)$, we can condense this to be: $$ \frac{\mathrm dy}{\mathrm dx} = \frac{\mathrm dy}{\mathrm du} \cdot \frac{\mathrm du}{\mathrm dx} $$