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use the chain rule to evaluate the derived function for
x
x
.
Extend this result to show that for any rational number
r
0, the
derived function of
g
, where
g
(
x
)
x
, for positive
x
, is given by
g
(
x
)
rx
.
For what rational numbers
r
is
g
differentiable at
x
0?
44 Find a maximal interval domain containing zero for which the sine
function has a well-defined inverse. Find the derived function of this
inverse function (known as arcsine) and statethemaximal intrval
domain for this derived function. You may use any of the
properties of the circular functions which are established in chapter
11.
45 Defining tan
x
(sin
x
)/(cos
x
), provided cos
x
0, identify a
maximal interval domain containing zero for which the tangent
function has a well-defined inverse. Find the derived function of this
inverse function (known as arctan) and statethemaximal domain
for this derived function. You may use the properties of the circular
functions established in chapter 11.
Derivatives at end points
Wesay that a function
f
is differentiable on a set
A
when
f
is
differentiable at each point of
A
.If
A
is an open interval, the meaning is
clear enough, since an open interval contains a neighbourhood of each
of its points. But functions such as
x
x
may be defined on [0,
)
and in such a caseit may beimportant to beableto aMrm or deny the
differentiability of thefunction at 0.
46 Look back to the definition of derivative and propose a criterion to
determine whether a function
f
:[
a
,
b
]
R is differentiable at
a
.
Modify your criterion to determine differentiability (or otherwise) at
b
.
Summary
Definition
qn 4
If
a
is a cluster point of the domain of a real
function
f
, then, when the limit
f
(
x
)
f
(
a
)
x
a
lim
exists,
f
is said to be
differentiable
at
a
and the
