At the moment I am facing a challenge concerning the exchange of an integral and a partial derivative. Not sure, if it is right, but is there any assumption that the following holds or what is the correct solution:
$\frac{\partial}{\partial f(x,t)}\int_a^bf(x,t)^2 -g(x,t)dt = \int_a^b 2f(x,t)dt$
I am facing two problems that might not be problems:
a) the derivative $\partial f(x,t)$ depends on the integral variable $dt$. This case is not covered in the Leibniz rule, as far as I understand. Can I still exchange the partial derivative and the integral?
b) If I derive $g(x,t)$ wrt to $f(x,t)$ is it really 1, although they depend on the same variables?
Help and any ideas are really much appreciated. Thank you!