SIFT About Difference-of-Gaussian function extrema?
You have to take the derivative with respect to the vector $x$ and set it equal to zero. For a constant matrix $A$, the derivative of $A^Tx$ is $A$, and the derivative of $\frac12 x^TA^Tx=Ax$. So taking the derivative of $(1)$ gives

$$\frac{\partial D}{\partial x}+\frac{\partial^2D}{\partial x^2}x\tag{1}$$

Setting $(1)$ equal to zero results in $(2)$.