# termwise differentiation

###### Theorem.

If in the open interval $I$, all the of the series

${f}_{1}(x)+{f}_{2}(x)+\mathrm{\cdots}$ | (1) |

have continuous^{} derivatives^{},
the series converges^{} having sum $S(x)$ and
the differentiated series ${f}_{1}^{\prime}(x)+{f}_{2}^{\prime}(x)+\mathrm{\cdots}$
converges uniformly (http://planetmath.org/SumFunctionOfSeries) on the interval $I$,
then the series (1) can be differentiated termwise, i.e. in every point of $I$ the sum function $S(x)$ is differentiable^{} and

$$\frac{dS(x)}{dx}={f}_{1}^{\prime}(x)+{f}_{2}^{\prime}(x)+\mathrm{\cdots}$$ |

The situation implies also that the series (1) converges uniformly on $I$.

Title | termwise differentiation |
---|---|

Canonical name | TermwiseDifferentiation |

Date of creation | 2013-03-22 14:38:38 |

Last modified on | 2013-03-22 14:38:38 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 9 |

Author | Mathprof (13753) |

Entry type | Theorem |

Classification | msc 26A15 |

Classification | msc 40A30 |

Synonym | differentiating a series |

Related topic | PowerSeries |

Related topic | IntegrationOfLaplaceTransformWithRespectToParameter |

Related topic | IntegralOfLimitFunction |