I have a bit of an unusual setup and would just like to know if the steps I am taking make sense. I have the following matrices, $A$ is an invertible $p \times p$ matrix, $B$ is a $p \times 1$ matrix or vector whichever wording you prefer, $D$ is a block-diagonal $p \times p$ matrix and not necessarily invertible, $\lambda$ is a constant/scalar, and $\sigma^2$ is the scalar of interest. With these parameters, I have the following defined function,
$$Q=[(A+\frac{\lambda}{\sigma^2}D)^{-1}B]^T$$ Where $T$ stands for the transpose. I now wish to take the derivative of this function with respect to $\sigma^2$. So I would just like to know since the $\sigma^2$ is not a matrix but a scalar what would be the next step I would take and does the transpose play a role in this at all?
$$\frac{dQ}{d \sigma^2}=\frac{d}{d \sigma^2}[(A+\frac{\lambda}{\sigma^2}D)^{-1}B]^T$$
Thank you