I-Calculus
9 - Power Rule for Differentiation (6)
Suppose
y
=
f
(
x
)
=
x
q
where
q
is an irrational number. And, suppose
{
r
k
}
k
=
1
k
=
∞
is an infinite sequence of rational numbers such that
lim
k
→
∞
r
k
=
q
. Then
d
y
d
x
=
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
=
lim
h
→
0
(
x
+
h
)
q
−
x
q
h
=
lim
h
→
0
(
x
+
h
)
lim
k
→
∞
r
k
−
x
lim
k
→
∞
r
k
h
=
lim
k
→
∞
lim
h
→
0
(
x
+
h
)
r
k
−
x
r
k
h
=
lim
k
→
∞
r
k
x
r
k
−
1
=
lim
k
→
∞
r
k
lim
k
→
∞
x
r
k
−
1
=
q
x
q
−
1
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