# 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  $t=t_{0}$.

Title derivative of Riemann integral DerivativeOfRiemannIntegral 2013-03-22 14:35:30 2013-03-22 14:35:30 PrimeFan (13766) PrimeFan (13766) 9 PrimeFan (13766) Theorem msc 26A24 msc 26A42 DifferentiationUnderIntegralSign