Product rule for matrix-vector product

Question

Suppose that $x \in L^2([0,1], \mathbb{R}^m)$ is a vector valued function and $A(x)$ is a ($m \times m$)-matrix whose entries are the components of $x$. Then consider a differentiable curve $\gamma: (-1,1) \to L^2([0,1], \mathbb{R}^m)$. Now I want to find an expression of the derivative $\frac{d}{dt} A(\gamma (t)) \gamma (t) $. By the chain rule we have $\frac{d}{dt} A(\gamma (t))=DA_\gamma (\gamma (t)))\frac{d}{dt} \gamma (t)$, where $DA_\gamma$ denotes the derivative of $A$ with respect to its argument. But I want to compute the derivative of the product $A(\gamma (t)) \gamma (t) $. So, my question is: Is there something like a product rule for these kind of products and how can I compute the derivative then? I would be thankful for help. Thanks in advance.