Derivative of vector and vector transpose product

Question

I saw this answer here : Vector derivative w.r.t its transpose $\frac{d(Ax)}{d(x^T)}$.

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I am finding difficult to understand the part in red. What rule is that ?

If I apply multiplication rule, shouldn't I get -

And how do one differentiate this anyways ?

Answer 1

You can do it componentwisely:

$$\frac{d(x^Ta)}{dx}=\left(\frac{d(x_1a_1+x_2a_2+\cdots+x_na_n)}{dx_1}, \frac{d(x_1a_1+x_2a_2+\cdots+x_na_n)}{dx_2}, \cdots, \frac{d(x_1a_1+x_2a_2+\cdots+x_na_n)}{dx_n}\right)\\ =(a_1,a_2,\cdots, a_n)=a^T$$