Taylor series operator commutes with differentiation

Statement

Suppose $f$ if a function defined on a subset of the reals that is infinitely differentiable at a point $x_0$ in its domain. Then, the derivative $f^{\prime }$ is also defined and infinitely differentiable at $x_0$, and the Taylor series for $f^{\prime }$ is the derivative (in the sense of derivative of power series) of the Taylor series for $f$.

Related facts

• Differentiation theorem for power series
• Taylor series operator is linear
• Taylor series operator is multiplicative