Graphics Reference
In-Depth Information
15 By using qns 10, 11 and 14 find
f
(
a
) when
f
(
x
)
b
b
x
b
x
...
b
x
.
16 If
f
(
x
)
1/
x
, provethat
f
(
a
)
n
/
a
, for any positive integer
n
,
provided
a
0.
The quotient rule
17 If two real functions
f
and
g
are both differentiable at
a
, prove
using qns 6.93 and 12 that thefunction
f
/
g
defined by
f
/
g
:
x
f
(
x
)/
g
(
x
) is differentiable at
a
, that
f
g
(
a
)
g
(
a
) ·
f
(
a
)
f
(
a
) ·
g
(
a
)
(
g
(
a
))
,
provided
g
(
a
)
0.
The chain rule
18 If thereal function
g
is differentiable at
a
, and thereal function
f
is
differentiable at
g
(
a
), we seek to prove that the composite function
f
g
defined by
f
g
:
x
f
(
g
(
x
)) is differentiable at
a
, and that
(
f
g
)
(
a
)
f
(
g
(
a
))·
g
(
a
).
(i) First suppose that there is a neighbourhood of
a
in which
g
(
x
)
g
(
a
), unless
x
a
, and usethecontinuity of
g
at
a
and
theequation
f
(
g
(
x
))
f
(
g
(
a
))
x
a
f
(
g
(
x
))
f
(
g
(
a
))
g
(
x
)
g
(
a
)
·
g
(
x
)
g
(
a
)
x
a
,
for
x
a
, to provethersult.
(ii) If in every neighbourhood of
a
there are points
x
such that
g
(
x
)
g
(
a
), the cancellation on which part (i) is based cannot
be carried out. For this case only, we construct an argument
establishing the chain rule by showing that each side of the
equation (
f
g
)
(
a
)
f
(
g
(
a
))·
g
(
a
)is0.
Suppose that in every neighbourhood of
a
there are points
x
a
for which
g
(
x
)
g
(
a
).
Let
A
x
g
(
x
)
g
(
a
),
x
a
and let
g
(
x
)
g
(
a
)
G
(
x
)
, for
x
a
.
x
a
Since
g
is differentiable at
a
, for any sequence (
a
)
a
with
a
a
, in thedomain of
g
,(
G
(
a
))
g
(
a
).
