$$u(x) = \int_0^1 xtu^2(t) dt$$
$u$ is an unknown function (integral equation form). If not possible to differentiate explicitly is there an expression for the derivative of this integral with respect to $u$?
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So you want to take derivative with respect to a constant? – user10354138 Sep 29 '23 at 04:25
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u is a function depending on x. Not a constant. I wonder if it is possible to differentiate with respect to u. – Log On Sep 29 '23 at 04:29
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1So you want the functional derivative $\frac{\delta u(x)}{\delta u(y)}$ but the function $u$ isn't allowed to vary, i.e., a constant. – user10354138 Sep 29 '23 at 04:31
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Do you really mean the derivative with respect to $u$, or with respect to $x$ (in other words $u'(x)$)? – Annie Carter Sep 29 '23 at 04:33
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I do mean with respect to u yes. – Log On Sep 29 '23 at 04:34
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You can't just differentiate an expression like that. The derivative of the left hand side is just the identity man (as with any linear function) and the derivative of the right hand side is the map $h \mapsto 2\int_0^1 x t u(t)h(t)dt$. Assuming that you are using appropriate spaces so all is well defined. – copper.hat Sep 29 '23 at 05:09