I have a function $J = \mathit{f}(A(a,b))$, that is, matrix A that is a function of parameters $a \ \& \ b$, for example $$A = \begin{bmatrix}1&2&3&4\\5&6&7&1\\a&a&a&0\\b&b&0&b\end{bmatrix}$$
I want to compute the derivatives $\frac{\partial J}{\partial a}$ and $\frac{\partial J}{\partial b}$. Given $\frac{\partial J}{\partial A}$, Is the derivative below correct?
I am doubting this since $\frac{\partial A}{\partial a}$ and $\frac{\partial A}{\partial b}$ is no longer an element-wise differential $$\frac{\partial J}{\partial a} = Tr(\left[\frac{\partial J}{\partial A}\right]^T \times \frac{\partial A}{\partial a})$$
$$\frac{\partial J}{\partial b} = Tr(\left[\frac{\partial J}{\partial A}\right]^T \times \frac{\partial A}{\partial b})$$
$$\frac{\partial A}{\partial a} = \begin{bmatrix}0&0&0&0\\0&0&0&0\\1&1&1&0\\0&0&0&0\end{bmatrix}$$