Computation of the derivative of a matrix-valued function

Question

Let $a,b,c \in \mathbb{N}$, $f:\mathbb{R}^a \rightarrow \mathbb{R}^{b \times c}$ be Frèchet-differentiable with Frèchet-derivative $Df$ and let $y,x \in \mathbb{R}^a$.

Is there a way to compute $(Df)(x)y$?

I know that $Df$ is the Jacobi matrix in case of b=a,c=1, but I have no idea how to compute $(Df)(x)y$ for general $a,b$. I think that $(Df)(x)y \in \mathbb{R}^{b \times c}$ holds.

I would be very grateful for any help.

Answer 1

$$ f(x_0) = \Big( \left.\frac{∂f_{}}{∂x_{}}\right\vert_{x=x_0} \Big)_{, } \qquad f(x_0)(y) = \Big(\sum_{} f(x_0)_{, } y_{}\Big)_{} $$

For multi-indices $$ and $$.