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Mathematics
Mathematics
Problem
(i) For a positive integer
n
n
n
and base
a
>
0
,
a
≠
1
a>0,\;a\neq1
a
>
0
,
a
=
1
, prove that
d
d
x
[
log
a
(
x
n
)
]
=
n
x
ln
a
,
x
>
0.
\frac{d}{dx}\bigl[\log_a\!(x^{n})\bigr]=\frac{n}{x\ln a}, \quad x>0.
d
x
d
[
lo
g
a
(
x
n
)
]
=
x
ln
a
n
,
x
>
0.
(ii) Using the result from part (i), find the derivative of
log
7
(
x
3
)
\log_7\!(x^3)
lo
g
7
(
x
3
)
at
x
=
2
x=2
x
=
2
.
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