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Thread: derivative with x occurring at both integrand and upper bound

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    derivative with x occurring at both integrand and upper bound

    If $\displaystyle f$ is continuously differentiable in $\displaystyle \mathbb R^2$, $\displaystyle g$ is continuously differentiable in $\displaystyle \mathbb R$, $\displaystyle a$ a constant real, how to calculate the derivative of $\displaystyle \int_a^{g(x)}f(x,t)dt$ with regard to $\displaystyle x$ (not by numerical methods)? Thanks.
    Last edited by zzzhhh; Nov 25th 2009 at 08:15 AM.
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    Quote Originally Posted by zzzhhh View Post
    If $\displaystyle f$ is continuously differentiable in $\displaystyle \mathbb R^2$, $\displaystyle g$ is continuously differentiable in $\displaystyle \mathbb R$, $\displaystyle a$ a constant real, how to calculate the derivative of $\displaystyle \int_a^{g(x)}f(x,t)dt$ with regard to $\displaystyle x$ (not by numerical methods)? Thanks.

    Use Leibnitz's Rule for differentiation under the integral sign:

    $\displaystyle \frac{\partial}{\partial x}\int\limits_a^{g(x)}f(x,t)\,dt=\int\limits_a^{g( x)}\frac{\partial f(x,t)}{\partial x}\,dx+f(x,g(x))\frac{\partial g(x)}{\partial x}-f(x,a)\frac{\partial a}{\partial x}$

    Tonio
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    Thank you, erudite Tonio.
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    After reading the proof of this general Leibniz integral rule, I have to stress that the conditions supporting the special case of Leibniz integral rule(*) must be preserved to make possible the interchange of the lim and int process in producing the integral part of the result when $\displaystyle \Delta x\to 0$, the most important of which is the continuity of the partial derivative.
    (*) $\displaystyle \frac{d}{dx}\int_a^b f(x,t)dt=\int_a^b \frac{\partial}{\partial x}f(x,t)dt$ where a,b are both constant.
    ps: My question comes from the proof of Poincare's lemma in baby Rudin, the proof that $\displaystyle F\in\mathcal C'(E)$ is omitted as usual in Th10.38, I think the general Leibniz integral rule is necessary to prove it which is not mentioned at all in this book. I hope it is not the author's original intention to leave such a huge gap for me to fill. Thanks to Tonio I finally made it.
    Last edited by zzzhhh; Nov 26th 2009 at 01:56 PM.
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