Difference between derivatives of matrices and derivatives of function

Question

As for derivatives of function:

$$\frac{\partial (x*x)}{\partial {x}} = \frac{\partial (x^2)}{\partial {x}} = 2x$$

But for derivatives of matrices, since $\frac{\partial (\boldsymbol{a}^{T}\boldsymbol{x})}{\partial \boldsymbol{x}}{} = \boldsymbol{a}$,

plug $\boldsymbol{a} = \boldsymbol{x}$ we get:

$$\frac{\partial (\boldsymbol{x}^{T}\boldsymbol{x})}{\partial \boldsymbol{x}} = \boldsymbol{x}$$

Why $\frac{\partial (\boldsymbol{x}^{T}\boldsymbol{x})}{\partial \boldsymbol{x}}{} \neq 2\boldsymbol{x}$ ?

Answer 1

The equation posted is wrong.

$\frac{\partial (\boldsymbol{x}^{T}\boldsymbol{x})}{\partial \boldsymbol{x}}{} = 2\boldsymbol{x}$ is true.

the problem is plugging $a = x$ is improper since $x$ is a variable while $a$ is a constant.

• derivatives of matrices: $\frac{\partial (\boldsymbol{x}^{T}\boldsymbol{x})}{\partial \boldsymbol{x}}{} = 2\boldsymbol{x}$

• derivatives of function: $\frac{\partial (x*x)}{\partial {x}} = 2x$

While

• derivatives of matrices: $\frac{\partial (\boldsymbol{a}^{T}\boldsymbol{x})}{\partial \boldsymbol{x}}{} = \boldsymbol{a}$

• derivatives of function: $\frac{\partial (a*x)}{\partial {x}} = a$