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how to derive $\int_0^1 G(t,s) e(s)ds$ with respect to $t$

Where $G(t,s)$ is a Green function and $e:(0,1)\rightarrow \mathbb{R}$ continuous and $e\in L(0,1)$

Please help me

Thank you

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up vote 1 down vote accepted

You can commute the derivative and the integral operators in this case. See here.

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$\frac{d}{dt}(\int_0^1 G(t,s)e(s) ds)=\int_0^1 \frac{\partial G(t,s)}{dt} e(s)ds $ it's right ? – Vrouvrou May 16 '13 at 7:20
yes, you are correct – Wolphram jonny May 16 '13 at 7:39

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