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value of the limit is denoted by
f
(
a
).
Theorem
qn 12
If a function
f
is differentiable at
a
, then
f
is
continuous at
a
.
Theorem
If functions
f
and
g
are both differentiable at
a
,
then
The sum rule
qn 10
(
f
g
)
(
a
)
f
(
a
)
g
(
a
),
The product rule
qn 13
(
f
·
g
)
(
a
)
f
(
a
)·
g
(
a
)
f
(
a
) ·
g
(
a
),
The quotient rule
qn 17
and (
f
/
g
)
(
a
)
(
f
(
a
) ·
g
(
a
)
f
(
a
) ·
g
(
a
))/(
g
(
a
))
,
provided
g
(
a
)
0.
Theorem
qn 15
If
f
(
x
)
b
b
x
b
x
...
b
x
,
then
f
b
a
a
nb
a
(
a
)
2
b
3
b
...
.
The chain rule
qn 18
If thefunction
g
is differentiable at
a
and the
function
f
is differentiable at
g
(
a
), then
(
f
g
)
(
a
)
f
(
g
(
a
))·
g
(
a
).
Definition
qn 29
A function
f
is said to havea
local maximum
at
a
if, for all
x
in somenighbourhood of
a
,
f
(
x
)
f
(
a
). A function
f
is said to havea
local
minimum
at
a
if for all
x
in somenighbourhood
of
a
,
f
(
x
)
f
(
a
).
Theorem
qn 30
If thefunction
f
has a local maximum or a local
minimum at
a
, and
f
is differentiable at
a
, then
f
(
a
)
0.
Definition
qns 36, 38
If
f
:
A
R is differentiable at each point of
B
A
, then
f
is called the
derived
function
or
derivative
of
f
.
By induction wedefinethe
n
th
derivative
of
f
,
denoted by
f
, as thedrivativeof
f
.
:
B
R
Theorem
qn 40
If
f
is a bijection and
g
is its inverse, and if
f
is
differentiable at
a
and
g
is differentiable at
f
(
a
),
then
g
(
f
(
a
))
1/
f
(
a
).
Historical note
From one point of view, most of the material of this chapter was
known by mathematicians before Newton's creative burst in 1664
—
6.
From another point of view, none of the results of this chapter were
proved in the sense in which we take them today until the time of
Weierstrass, 1860
—
70.
The essential backcloth to the calculus is Descartes' method of
converting geometric problems into algebraic ones (coordinate
