Derivative of a multivariate quadratic

Question

What is the derivative of vector transpose?

for example, if I want to minimize function which is written like this $\frac12 x^Tx-b^Tx$ subject to $x\in\Bbb R^n$. What will be the derivative of this function?

Answer 1

The function is $$F = \frac12 \boldsymbol{x}^T\boldsymbol{x}-\boldsymbol{b}^T\boldsymbol{x}.$$

The total derivative with respect to $\boldsymbol{x}$ is given as

$$dF = \frac12 d\boldsymbol{x}^T\boldsymbol{x}+\frac12 \boldsymbol{x}^Td\boldsymbol{x}-\boldsymbol{b}^Td\boldsymbol{x}$$ $$\implies dF = d\boldsymbol{x}^T\left[\boldsymbol{x}-\boldsymbol{b} \right].$$

A $dF = d\boldsymbol{x}^T\text{grad}F$ comparison yields $$\text{grad}F = \boldsymbol{x}-\boldsymbol{b}.$$