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and
2
4
3
5
q
f (x)
q
x
1
q
f (x)
q
x
2
.
q
f (x)
q
x
n
2
4
3
5
c
1
c
2
.
c
n
q
f (x)
q
x
¼
¼
Hence,
q
c
T
x
q
x
¼ c
(3
:
208)
Similarly,
q
x
T
c
q
x
¼ c
(3
:
209)
3.12.2 D
ERIVATIVES OF
Q
UADRATIC
F
UNCTIONS
Let x
2
R
n
and f
(
x
) ¼
x
T
Ax, where A is an n
n symmetric matrix. The multivariate
quadratic function f
(
x
)
can be written as
X
X
n
n
f
(
x
) ¼
x
T
Ax
¼
a
ij
x
i
x
j
(
3
:
210
)
i
¼
1
j
¼
1
Consider the first component of vector
q
f
(
x
)
q
x
, that is,
q
x
1
X
X
q
x
1
X
X
n
n
n
n
q
f
(
x
)
q
x
1
¼
q
a
ij
x
i
x
j
¼
q
a
ij
x
i
x
j
þ
q
q
x
1
a
11
x
1
x
1
i
¼
1
j
¼
1
i
¼
2
j
¼
2
q
x
1
X
q
x
1
X
n
n
þ
q
a
1j
x
1
x
j
þ
q
a
i1
x
i
x
1
j
¼
2
i
¼
2
X
X
q
x
1
x
1
X
q
x
1
x
1
X
n
n
n
n
¼
q
q
x
1
a
ij
x
i
x
j
þ
q
q
x
1
a
11
x
1
þ
q
a
1j
x
j
þ
q
(
:
)
a
i1
x
i
3
211
i
¼
2
j
¼
2
j
¼
2
i
¼
2
The
first term in the above expansion is independent of x
1
, therefore it does not
contribute to the derivative. Hence,
X
n
X
n
X
n
X
n
2
X
n
q
f
(
x
)
q
x
1
¼
0
þ
2a
11
x
1
þ
a
1j
x
j
þ
a
i1
x
i
¼
a
1j
x
j
þ
a
i1
x
i
¼
a
1j
x
j
j
¼
2
i
¼
2
j
¼
1
i
¼
1
j
¼
1
(
3
:
212
)
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