Matricial differentiation $x x^{\top} b $

Question

What is the drivative of $x x^{\top} b $ with respect to x, knowing that b is constant vector?

Answer 1

Let $f(x) = x x^T b$, then $f(x+h) = (x+h)(x+h)^T b = f(x)+h x^T b+ x h^T b + h h^Tb$. Since $\|h h^T b\| \le \|h\|^2 \|b\|$, we see that $Df(x)(h) = h x^T b+ x h^T b$.

Since $h x^T b = x^T b h$ and $x h^T b = x b^T h$, we have $Df(x)(h) = (x^T b I + x b^T) h$, so we can write $Df(x) = x^T b I + x b^T$.

Answer 2

Let $v = xx'\cdot b$, then find the differential
$$ \eqalign { dv &= (dx)x'\cdot b + x(dx')\cdot b \cr &= (I\cdot dx)(x'\cdot b) + x(b'\cdot dx) \cr &= \big[(x'\cdot b) I + x\,b'\big]\cdot dx \cr } $$ So the derivative must be $$ \eqalign { \frac {\partial v} {\partial x} &= (x'\cdot b) I + x\,b' \cr } $$