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Homederivative of Riemann integral

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# 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 $f^{{\prime}}_{t}(x,\,t)$ 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

$F(t)=\int_{{a(t)}}^{{b(t)}}f(x,\,t)\,dx$ |

and

$G(t)=b^{{\prime}}(t_{0})\cdot f(b(t),\,t)-a^{{\prime}}(t_{0})\cdot f(a(t),\,t)% +\int_{{a(t)}}^{{b(t)}}f^{{\prime}}_{t}(x,\,t)\,dx.$ |

Then one has the derivative

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

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

Related:

DifferentiationUnderIntegralSign

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

26A24*no label found*26A42

*no label found*

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