课堂上老师介绍的几个求偏导的公式，但是不知道为什么是这么个结果，只有课下带入实例计算一下才能更好的理解。
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(frac{partial eta^{mathrm{T}} mathrm{x}}{partial mathrm{x}}=eta)
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(frac{partial mathrm{x}^{mathrm{T}} mathrm{x}}{partial mathrm{x}}=2 mathrm{x})
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(frac{partial mathrm{x}^{mathrm{T}} mathrm{Ax}}{partial mathrm{x}}=left(mathrm{A}+mathrm{A}^{mathrm{T}} ight) mathrm{x})
对于上述三个求导公式，通过带入实例进行求导计算，令：
[eta =
egin{bmatrix}
eta_1 \
eta_2 \
eta_3
end{bmatrix}\
mathrm{x}=
egin{bmatrix}
x_1 \
x_2 \
x_3
end{bmatrix}\
A =
egin{bmatrix}
a_{11} & a_{12} & a_{13}\
a_{21} & a_{22} & a_{23} \
a_{31} & a_{32} & a_{33}
end{bmatrix}
]
第一个公式
[eta^Tmathrm{x} = egin{bmatrix}
eta_1 &
eta_2 &
eta_3
end{bmatrix}
egin{bmatrix}
x_1 \
x_2 \
x_3
end{bmatrix}
=eta_1x_1+eta_2x_2+eta_3x_3\
\
frac{partial eta^{mathrm{T}} mathrm{x}}{partial mathrm{x}} = egin{bmatrix}
frac{partial (eta_1x_1+eta_2x_2+eta_3x_3)}{partial x_1} \
frac{partial (eta_1x_1+eta_2x_2+eta_3x_3)}{partial x_2} \
frac{partial (eta_1x_1+eta_2x_2+eta_3x_3)}{partial x_3}
end{bmatrix}
=
egin{bmatrix}
eta_1 \
eta_2 \
eta_3
end{bmatrix}
=
eta
]
第二个公式
[mathrm{x}^Tmathrm{x} =
egin{bmatrix}
x_1&x_2&x_3
end{bmatrix}
egin{bmatrix}
x_1 \
x_2 \
x_3
end{bmatrix}
=
x_1^2+x_2^2+x_3^2\
frac{partial mathrm{x}^{mathrm{T}} mathrm{x}}{partial mathrm{x}}
=
egin{bmatrix}
frac{partial (x_1^2+x_2^2+x_3^2)}{partial x1} \
frac{partial (x_1^2+x_2^2+x_3^2)}{partial x2} \
frac{partial (x_1^2+x_2^2+x_3^2)}{partial x3}
end{bmatrix}
=
egin{bmatrix}
2x_1 \
2x_2 \
2x_3
end{bmatrix}
=
2
egin{bmatrix}
x_1 \
x_2 \
x_3
end{bmatrix}
=
2mathrm{x}
]
第三个公式
[egin{aligned}
mathrm{x}^{T} A mathrm{x} &=left[egin{array}{lll}
x_{1} & x_{2} & x_{3}
end{array}
ight]left[egin{array}{lll}
a_{11} & a_{12} & a_{13} \
a_{21} & a_{22} & a_{23} \
a_{31} & a_{32} & a_{33}
end{array}
ight]left[egin{array}{l}
x_{1} \
x_{2} \
x_{3}
end{array}
ight] \
&=a_{11} x_{1}^{2}+a_{21} x_{2} x_{1}+a_{31} x_{3} x_{1}+a_{12} x_{1} x_{2}+a_{22} x_{2}^{2}+a_{32} x_{3} x_{2}+a_{13} x_{1} x_{3}+a_{23} x_{2} x_{3}+a_{33} x_{3}^{2}
end{aligned}\
]
[egin{aligned}frac{partial mathrm{x}^{mathrm{T}} mathrm{Ax}}{partial mathrm{x}}=&left[egin{array}{l}frac{partial mathrm{x}^{mathrm{T}} mathrm{Ax}}{partial x_{1}} \frac{partial mathrm{x}^{mathrm{T}} mathrm{Ax}}{partial x_{2}} \frac{partial mathrm{x}^{mathrm{T}} mathrm{Ax}}{partial x_{3}}end{array}
ight] \=&left[egin{array}{l}2 a_{11} x_{1}+a_{21} x_{2}+a_{31} x_{3}+a_{12} x_{2}+a_{13} x_{3} \a_{21} x_{1}+a_{12} x_{1}+2 a_{22} x_{2}+a_{33} x_{3}+a_{23} x_{3} \a_{31} x_{1}+a_{32} x_{2}+a_{13} x_{1}+a_{23} x_{2}+2 a_{33} x_{3}end{array}
ight] \=&left[egin{array}{l}2 a_{11} x_{1}+left(a_{21}+a_{12}
ight) x_{2}+left(a_{31}+a_{13}
ight) x_{3} \left(a_{21}+a_{12}
ight) x_{1}+2 a_{22} x_{2}+left(a_{33}+a_{23}
ight) x_{3} \left(a_{31}+a_{13}
ight) x_{1}+left(a_{32}+a_{23}
ight) x_{2}+2 a_{33} x_{3}end{array}
ight] \=&left[egin{array}{l}2 a_{11}+left(a_{21}+a_{12}
ight)+left(a_{31}+a_{13}
ight) \left(a_{21}+a_{12}
ight)+2 a_{22}+left(a_{33}+a_{23}
ight) \left(a_{31}+a_{13}
ight)+left(a_{32}+a_{23}
ight)+2 a_{33}end{array}
ight]left[egin{array}{l}x_{1} \x_{2} \x_{3}end{array}
ight] \=&left(A+A^{T}
ight) mathrm{x}end{aligned}
]