I have a function as following
$$F=\int |A^TG(x)-B^TJ(x)|^2 H(x)\,dx+ \int |A^TG(x)-C^TJ(x)|^2 (1-H(x)) \, dx+\lambda_1 A^2+\lambda_2 B^2+\lambda_2 C^2$$
where $A^T$ is transpose of vector $A$. $A$ is a column vector such as $A= \begin{bmatrix} a_1 \\ a_2 \\ ... \\ a_n \\ \end{bmatrix} $; $B= \begin{bmatrix} b_1 \\ b_2 \\ ... \\ b_n \\ \end{bmatrix} $; $C= \begin{bmatrix} c_1 \\ c_2 \\ ... \\ c_n \\ \end{bmatrix} $; $A^TG(x)=\sum_{i=1}^{i=n} a_i G_i $
$G=[G_1...G_i..G_n],J=[J_1...J_i..J_n]$, in which $G_i$ is matrix $n\times n$ and $H(x)$ is Heaviside smoothing function. Note that given $G,J,H$
I want to find the coefficients of $A,B$ and $C$ subject to $\min F$. Hence, we have equation.
$$\frac {\partial F}{\partial A}=0$$ $$\frac {\partial F}{\partial B}=0$$ $$\frac {\partial F}{\partial C}=0$$ However, How to find the derivative of these above function?
$$\frac {\partial F}{\partial A}=?$$ $$\frac {\partial F}{\partial B}=?$$ $$\frac {\partial F}{\partial C}=?$$