This is my little attempt to visualize the nested composition of functions:

func = Cos[Log[x^2 + 1]]^3;
func //. Cases[func, f_[g_] :> (f[g] :> f[Framed[g]]), All]
% //. Cases[%, f_^n_Integer /; f =!= x :> (f^n :> Framed[f]^n), All]

Michael, thank you for such a detailed and insightful response! Your point about the "extreme elaborateness" of complex branching functions is a powerful argument for the Outside-In approach. You are absolutely right—once you introduce products or sums within the composition, like in your example $f(g(h(k)\cdot m(n+p(q))))$, the "inside" becomes a moving target. Starting from the outermost operator is the only way to maintain a consistent algorithmic path. I particularly like your use of color-coding to track the "active" differentiation layer. In developing

Derivative Calculus Solver

, I faced a similar challenge in UI design: how to show the "Inside" function clearly while the "Outside" is being processed. Inspired by the hybrid Leibniz/Differential method you mentioned ( $d(f(u)) = f'(u) du$), I structured my tool to explicitly define $u$ at each step. This seems to help students who struggle with the "bookkeeping" aspect of the chain rule that you highlighted. Your custom Wolfram code for the dt function is excellent. The way it recursively generates the "Step Tree" using ColorData is a brilliant pedagogical tool for the community. I will certainly be spending some time studying your to Univariate Power rules—that’s a very elegant way to handle Power[] objects for visualization. Thank you for sharing your 1981-to-present perspective; it’s rare to see such a clear bridge between the history of CAS and modern teaching methods!