For $g: R^{n \times p} \to R^{n \times n}, U \mapsto UU^{T}$ what is the full derivative of $g$ respect to $U$
This questions arises when I am trying to find the derivative of $\langle C,UU^{T} \rangle$, with respect to the matrix U, where C is a constant $n \times n$ matrix and the inner product is canonical. I attempted to use the chain rule and got the above question
I have spent quite some time searching up this question, and I did find many posts that seems to be related. However, as I don't have much knowledge in differential geometry nor machine learning, I quickly become overwhelmed to the notation. I am doing this problem with the full derivative so I expect the solution to be a $nn \times np$ matrix, but if there are other nicer notations to use, please let me know as well.
Just out of curiosity, what can I do in general to find the derivative of $f: X \mapsto u(X)v(X)$ where $X$ is a matrix and $u(X)$ and $v(X)$ outputs a matrix