derivative of Riemann integral

Let $f$ be a continuous function from an open subset $A$ of ${ℝ}^2$ to $ℝ$.  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

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

and

$$G(t)=b^{\prime }(t_0)\cdot f(b(t),t)-a^{\prime }(t_0)\cdot f(a(t),t)+{\int }_{a(t)}^{b(t)}{f}_t^{\prime }(x,t)𝑑x.$$

Then one has the derivative

$$F^{\prime }(t_0)=G(t_0)$$

in all such points  $t=t_0$.

Title	derivative of Riemann integral
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