I'm stuck on the following exercise:
Let $a \in \mathbb{R}^{n \times n}$ and $b \in R^n$. We consider the following >function:
$$f\colon x \in \mathbb{R}^n \mapsto \frac{1}{2} x \cdot Ax + b \cdot x$$
Calculate the first and second derivative of $f$.
My idea would be to use the rule for the Derivative of the dot product (as seen here):
Let
$$\mathbf r\colon x \mapsto \left\langle{r_1 \left({x}\right), r_2 \left({x}\right), \dots, r_n \left({x}\right)}\right\rangle$$
$$\mathbf q\colon x \mapsto \left\langle{q_1 \left({x}\right), q_2 \left({x}\right), \dots, q_n \left({x}\right)}\right\rangle$$
It holds:
$$\dfrac d {d x} \left({\mathbf r \left({x}\right) \cdot \mathbf q \left({x}\right)}\right) = \mathbf r' \left({x}\right) \cdot \mathbf q \left({x}\right) + \mathbf r \left({x}\right) \cdot \mathbf q' \left({x}\right)$$
But the presence of the matrix $A$ in $f$ confuses me. Could you help me?