Let $\mathbf{x}\in\Bbb{R}^n$, $\mathbf{y}\in\Bbb{R}^m$, and $A\in\Bbb{R}^{m\times n}$. Also, let $f\colon\Bbb{R}^n\to\Bbb{R}$ given by $$ f(\mathbf{x}) = \big(A\mathbf{x}\big)\cdot\mathbf{y} = \big(A\mathbf{x}\big)^\top\mathbf{y} $$ What is the first derivative of $f$ (which will belong to $\Bbb{R}^n$) with respect to $\mathbf{x}$, i.e., $$ \frac{\partial f}{\partial\mathbf{x}}=\frac{\partial(A\mathbf{x})^\top\mathbf{y}}{\partial\mathbf{x}}=? $$
Will it be just $$ \frac{\partial(A\mathbf{x})^\top\mathbf{y}}{\partial\mathbf{x}} = A^\top\mathbf{y}, $$
or something else? Thanks a lot in advance!
Looks right as for vectors over $\mathbb{R}^j$,
$$Ax \cdot y = y \cdot Ax = y^TAx = (A^Ty)^T x$$