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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.

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Yes, the standard product rule applies. The gradient of the product is $$f(X)\nabla_X g(X)+g(X)\nabla_X f(X).$$ The dimensions of the gradients, of course, are the same as those of $X$ itself.

You might find The Matrix Cookbook useful here.

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The product rule holds in very great generality. Let $X,Y,Z,W$ be Banach spaces with open subset $U \subset X$, and suppose $f: U \rightarrow Y$ and $g: U \rightarrow Z$ are Frechet differentiable. If $B(\cdot, \cdot): Y \times Z \rightarrow W$ is a continuous bilinear map, then

$$ \frac{d}{dx}[ B(f(x), g(x))] = B(f'(x), g(x)) + B(f(x), g'(x))$$

where all the derivatives in question are Frechet derivatives. To apply to your case, we take $U = X = \mathbb{R}^{n \times n}$, $Y =Z = W = \mathbb{R}$, and $B(y,z) = yz$.

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