I have to find the derivative of a function. Could you help me to find it $$F(u)=\sum_{i=1}^{N} \int_{\Omega} f^2(x) u_i^q(x) dx $$
where $q \ge 1$, $f(x): \Omega \to R$, $u_i$ is membership function in which $u_i(x)=1$ for $x \in \Omega$, $0 \le u_i(x) \le 1, \sum_i u_i=1$
How to find: $$\frac {\partial F(u)} {\partial u}=?$$
Note that we need to consider 2 cases: $q=1$ and $q>1$ As my solution that I found that if
$$\frac {\partial F(u)} {\partial u}=0$$ Then solution of $u$ is $$u= \frac {(f^2(x))^{\frac {1}{1-q}}}{\sum_{i=1}^{N} (f^2(x))^{\frac {1}{1-q}}}$$
Thanks in advance