Taylor series operator commutes with composition

Statement

Suppose $x_{0}$ is a real number. Suppose $g$ is a function defined on a subset of the reals that is infinitely differentiable at $x_{0}$. Suppose $f$ is a function defined on a subset of the reals that is infinitely differentiable at $g(x_{0})$. Then, the composite of two functions $f\circ g$ is infinitely differentiable at $x_{0}$, and its Taylor series can be computed formally by composing the Taylor series for $f$ at $g(x_{0})$ with the Taylor series for $g$ at $x_{0}$. Formally, what this means is that we write down the Taylor series for $f$ at $g(x_{0})$, then plug in for $x$ the entire expression for the Taylor series of $g$, then simplify.

Related facts

Similar facts

• Taylor series operator is linear
• Taylor series operator is multiplicative
• Taylor series operator commutes with differentiation