# derivative of Riemann integral

Let $f$ be a continuous function^{} from an open subset $A$ of ${\mathbb{R}}^{2}$ to $\mathbb{R}$. Suppose that also the partial derivative^{} ${f}_{t}^{\prime}(x,t)$ is continuous in $A$ which contains the line segments along which the integration is performed and that $a(t)$ and $b(t)$ are real functions differentiable^{} in some point ${t}_{0}$. Denote

$$F(t)={\int}_{a(t)}^{b(t)}f(x,t)\mathit{d}x$$ |

and

$$G(t)={b}^{\prime}({t}_{0})\cdot f(b(t),t)-{a}^{\prime}({t}_{0})\cdot f(a(t),t)+{\int}_{a(t)}^{b(t)}{f}_{t}^{\prime}(x,t)\mathit{d}x.$$ |

Then one has the derivative^{}

$${F}^{\prime}({t}_{0})=G({t}_{0})$$ |

in all such points $t={t}_{0}$.

Title | derivative of Riemann integral |
---|---|

Canonical name | DerivativeOfRiemannIntegral |

Date of creation | 2013-03-22 14:35:30 |

Last modified on | 2013-03-22 14:35:30 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 9 |

Author | PrimeFan (13766) |

Entry type | Theorem |

Classification | msc 26A24 |

Classification | msc 26A42 |

Related topic | DifferentiationUnderIntegralSign |