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Unformatted text preview: directional derivatives of f at x0 exist and
!f
(x0 ) = dfx0 (v) ,
!v
(ii) for every direction v,
N # !f
!f
(x0 ) =
(x0 ) vi .
!v
! xi
i=1 (1) Proof. Since x0 is an interior point, there exists B (x0 , r) $ E . Let v # RN be
a direction and x = x0 + tv. Note that for t < r, we have that
&x % x0 & = &x0 + tv % x0 & = &tv& = t &v& = t 1 < r
and so x0 + tv # B (x0 , r) $ E . Moreover, x " x0 as t " 0 and so, since f is
di!erentiable at x0 ,
f (x0 + tv) % f (x0 ) % T (x0 + tv % x0 )
f (x) % f (x0 ) % T (x % x0 )
= lim
t!0
&x % x0 &
&x0 + tv % x0 &
f (x0 + tv) % f (x0 ) % tT (v)
= lim
.
t!0
t 0 = lim x!x0 Since
lim t!0 t
t is bounded by one, it follows that f (x0 + tv) % f (x0 ) % tT (v)
t f (x0 + tv) % f (x0 ) % tT (v)
= lim
= 0.
t!0 t
t
t But then
0 = lim t!0 f (x0 + tv) % f (x0 ) % tT (v)
f (x0 + tv) % f (x0 )
= lim
% T (v) ,
t!0
t
t
7 !f
which shows that there exists ! v (x0 ) = T (v).
!N
Part (ii) follows from the linearity of T . Indeed, writing v = i=1 vi ei , by
the linearity of T ,
"N
$
N
N
#
#
# !f
!f
(x0 ) = T (v) = T
vi ei =
vi T (ei ) =
(x0 ) vi .
!v
! xi
i=1
i=1
i=1 Remark 11 If in the previous theorem x0 is not an interior point but for some
direction v # RN , the point x0 is an accumulation point of the set E ) L, where
L is the line through x0 in the direction v, then as in the ﬁrst part of the proof
!f
we can show that there exists the directional derivative ! v (x0 ) and
!f
(x0 ) = dfx0 (v) .
!v 8...
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This document was uploaded on 03/31/2014.
 Spring '14
 Derivative

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