Given a commutative ring $R$, write $D_n(R)$ for the commutative ring $R[\varepsilon]/\varepsilon^n.$
In this notation, the taking of degree $n$ Taylor polynomials can be viewed as a function $$T_n : \mathcal{C}^\infty(\mathbb{R}) \rightarrow D_n(\mathcal{C}^\infty(\mathbb{R})), \qquad f \mapsto \sum_{i = 0}^n \frac{f^{(i)}}{i!}\varepsilon^i.$$
For example: $$T_2(x^5) = x^5+5x^4 \varepsilon+10x^3\varepsilon^2$$
It's well known that $T_n$ is an $\mathbb{R}$-algebra homomorphism; linearity is obvious, and the fact that $T_n$ preserves products is a fundamental consequence of the product rule. Long story short, I was wondering
Question. Is there's a useful formula for the $n$th-order Taylor polynomial of a composite? As in: $$T_n(g \circ f) \;= \;\cdots$$
I've been stuffing around with this for about half an hour or so, but to little avail.