I noticed this notation while going through a tutorial on matrix calculus: $$ \frac{\partial x^TAx}{\partial xx^T} = \frac{\partial}{\partial x}\left( \frac{\partial x^TAx}{\partial x} \right) = A^T+A $$
I'm not sure how the first equality is established (is the partial derivative of $x^TAx$ with respect to $xx^T$ actually equal to the second partial? if so can somebody plz show me). If the first term is simply a shorthand notation, wouldn't $ \frac{\partial^2 x^TAx}{\partial x\partial x^T} $ or $ \frac{\partial^2 x^TAx}{\partial x^2} $make more sense?
I'm aware of the notation $ \frac{\partial^2 f}{\partial x^2}$ in ordinary calculus, so I don't understand why the numerator and denominator of the shorthand notation above each uses only a single partial symbol...