# rules of calculus for derivative of formal power series

In this entry, we will show that the rules of calculus
hold for derivatives of formal power series. While
this could be verified directly in a manner analogous
to what was done for polynomials^{} in the parent entry,
we will take a different tack, deriving the results
for power series from the corresponding results for
polynomials. The basis for our approach is the
observation that the ring of formal power series can
be expressed as a limit of quotients of the ring of
polynomials:

$$A[[x]]=\underset{k\to \mathrm{\infty}}{lim}A[x]/\u27e8{x}^{k}\u27e9$$ |

Thus, we will proceed in two steps, first extending the
derivative operation^{} to the quotient rings^{} and showing that
its properties still hold there, then extending it to the
limit and showing that its properties hold there as well.

We begin by noting that the derivative is well-defined as a map from $A[x]/\u27e8{x}^{k+1}\u27e9$ to $A[x]/\u27e8{x}^{k}\u27e9$ for all integers $k\ge 0$.

###### Theorem 1.

Suppose that $A$ is a commutative ring, $k$ is a non-negative integer, and that $p$ and $q$ are elements of $A\mathit{}\mathrm{[}x\mathrm{]}$ such that $p\mathrm{\equiv}q$ modulo ${x}^{k\mathrm{+}\mathrm{1}}$. Then ${p}^{\mathrm{\prime}}\mathrm{\equiv}{q}^{\mathrm{\prime}}$ modulo ${x}^{k}$.

###### Proof.

By definition of congruence^{}, $p(x)=q(x)+{x}^{k+1}r(x)$ for
some polynomial $r\in A[x]$. Taking derivaitves, ${p}^{\prime}(x)={q}^{\prime}(x)+{x}^{k}(kr(x)+x{r}^{\prime}(x))$, so ${p}^{\prime}$ and ${q}^{\prime}$ are
equivalent^{} modulo ${x}^{k}$.
∎

It is easy to verify that the sum and product rules hold in this new context:

###### Theorem 2.

If $A$ is a commutative ring, $k$ is a non-negative integer, and $f\mathrm{,}g$ are elements of $A\mathit{}\mathrm{[}x\mathrm{]}\mathrm{/}\mathrm{\u27e8}{x}^{k\mathrm{+}\mathrm{1}}\mathrm{\u27e9}$, then ${\mathrm{(}f\mathrm{+}g\mathrm{)}}^{\mathrm{\prime}}\mathrm{=}{f}^{\mathrm{\prime}}\mathrm{+}{g}^{\mathrm{\prime}}$.

###### Proof.

Let $p,q$ be representatives of the equivalence classes^{}
$f,g$. Then we have ${(p+q)}^{\prime}={p}^{\prime}+{q}^{\prime}$ by the corresponding
theorem for polynomials. Hence, by definition of quotient,
we have ${(f+g)}^{\prime}={f}^{\prime}+{g}^{\prime}$.
∎

###### Theorem 3.

If $A$ is a commutative ring, $k$ is a non-negative integer, and $f\mathrm{,}g$ are elements of $A\mathit{}\mathrm{[}x\mathrm{]}\mathrm{/}\mathrm{\u27e8}{x}^{k\mathrm{+}\mathrm{1}}\mathrm{\u27e9}$, then ${\mathrm{(}f\mathrm{\cdot}g\mathrm{)}}^{\mathrm{\prime}}\mathrm{=}{f}^{\mathrm{\prime}}\mathrm{\cdot}g\mathrm{+}f\mathrm{\cdot}{g}^{\mathrm{\prime}}$.

###### Proof.

Let $p,q$ be representatives of the equivalence classes $f,g$. Then we have ${(p\cdot q)}^{\prime}={p}^{\prime}\cdot q+p\cdot {q}^{\prime}$ by the corresponding theorem for polynomials. Hence, by definition of quotient, we have ${(f\cdot g)}^{\prime}={f}^{\prime}\cdot g+f\cdot {g}^{\prime}$. ∎

When considering the chain rule, we need to note that composition does not always pass to the quotient, so we need to restrict the operands to obtain a well-defined operation. In particular, we will consider the following two cases:

###### Theorem 4.

If $A$ is a commutative ring, $p\mathrm{,}q\mathrm{,}r$ is a n element of $A\mathit{}\mathrm{[}x\mathrm{]}$, and $q\mathrm{\equiv}r$ modulo ${x}^{k}$ for some integer $k\mathrm{>}\mathrm{0}$, then $p\mathrm{\circ}q\mathrm{\equiv}p\mathrm{\circ}r$ modulo ${x}^{k}$.

###### Theorem 5.

If $A$ is a commutative ring, $k$ is a non-negative integer, and $p\mathrm{,}q\mathrm{,}P\mathrm{,}Q$ are elements of $A\mathit{}\mathrm{[}x\mathrm{]}$ such that $p\mathrm{\equiv}P$ modulo ${x}^{k}$, $q\mathrm{\equiv}Q$ modulo ${x}^{k}$ and $p\mathit{}\mathrm{(}\mathrm{0}\mathrm{)}\mathrm{=}\mathrm{0}$, then $p\mathrm{\circ}q\mathrm{\equiv}P\mathrm{\circ}Q$ modulo ${x}^{k}$.

[More to come]

Title | rules of calculus for derivative of formal power series |
---|---|

Canonical name | RulesOfCalculusForDerivativeOfFormalPowerSeries |

Date of creation | 2013-03-22 18:22:35 |

Last modified on | 2013-03-22 18:22:35 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 8 |

Author | rspuzio (6075) |

Entry type | Derivation^{} |

Classification | msc 12E05 |

Classification | msc 11C08 |

Classification | msc 13P05 |

Related topic | InvertibleFormalPowerSeries |