How do I find the minimum of this function? This might seem trivial to some of you, but I can't for the life of me figure out how to solve this.
$$\underset{x}\arg \min  (x - b)^T Ax$$
$$x \in \mathbb{R^n}$$
We may assume A to be invertable, but it is not symmetric.
My idea was to calculate the first and second derivative.
I know that $\frac{dx^T}{dx} = (\frac{dx}{dx})^T$, but when I try to apply the chain rule, I get 
$$\frac{d}{dx} = Ax + (x-b)^Tx$$
which doesn't make sense, as it's a vector plus a scalar.
Even if there is another way to find the x for which the function is minimal, I am now more interested in how to derive this kind of formula.

You can rewrite it to a standard quadratic program and use corresponding methods as follows:
$(x-b)^T A x = x^T A x - b^T A x = x^T A x - c^T x$
for $c := A^T b$.
Your method can work too but your derivative calculation was wrong, it would be:
$ \frac{d}{dx} (x-b)^T A x = (x-b)^T A + x^T A = 0 \Leftrightarrow 2A^T x = A^Tb $
