Jacobian of Dot Product - Matrices seem incorrect?

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I have the following expression

$m=A.b$ where each variable has a term $x$ inside, and these are the dimensions

$x$ - [1x1]

$b$ - [100x1]

$A$ - [100x100]

$m$ - [100x1]

I want to get the jacobian of $\frac{dm}{dx}$ which I assume is

$\frac{dm}{dx}=\frac{dA}{dx}.b+A.\frac{db}{dx}$

[100.1 x 1] = [100.100 x 1][1 x 100] + [100 x 100][100 x 1]

[100 x 1] = [10000 x 100] + [100 x 1] - Doesnt work

Unfortunately this doesnt add up as seen above. I am clearly doing something fundamentally wrong. Can someone please help me