Given a real rectangular matrix $X$, and two scalar-valued matrix functions, $f(X)$ and $g(X)$, does the product rule for differentiation of a product of scalar valued functions, hold when differentiating the product, $f(X)g(X)$ w.r.t $X$ ? If not, what would be the corresponding product rule? Let's assume that the product $f(X)g(X)$ gives a real valued scalar, and is well-defined in terms of the dimensions.
Note: $f(.)$ and $g(.)$ can be the matrix trace function for example.