# derivative of a function given as an integral with kernel

If the function $f(u)(x)$ is given as $f(u) = \int_{\Gamma} g(x,y) u(y) dy$, what is the derivative $df/du$?

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## migrated from mathoverflow.netJun 29 '13 at 2:01

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Formally, your function $f$ is linear in $u$. Do you know what is the derivative of a (continuous) linear function? – abatkai Jun 28 '13 at 19:50
Perhaps math.SE would be a better site for the question. – Igor Khavkine Jun 28 '13 at 22:35

The Gâteaux derivative of $f$ at $u$ along $v$ is $$Df_u(v) = f(v).$$ Note that the derivative does not depend on $u$. Hence it is also the Fréchet derivative of $f$. Note that we are implicitly assuming that the kernel $g$ is so that $f$ is a bounded linear map between the relevant function spaces you are interested in.