Find the derivative of log19 x
Answers
Step-by-step explanation:
Suppose we have
log
a
(
b
)
, we want to change it on exponential (e) base, then it can be written as:
log
a
(
b
)
=
log
a
(
e
)
⋅
log
e
(
b
)
Similarly, function
log
10
(
x
)
can be written as:
y
=
log
10
(
e
)
⋅
log
e
(
x
)
Let's say we have,
y
=
c
⋅
f
(
x
)
, where c is a constant
then,
y
'
=
c
⋅
f
'
(
x
)
Now, this is quite straightforward to differentiate, as
log
10
(
e
)
is constant, so only remaining function is
log
e
(
x
)
Hence:
y
'
=
log
10
(
e
)
⋅
1
x
Alternate solution:
Another common approach is to use the change of base formula, which says that:
log
a
(
b
)
=
ln
(
b
)
ln
(
a
)
From change of base we have
log
10
(
x
)
=
log
10
(
x
)
=
ln
(
x
)
ln
(
10
)
.
This we can differentiate as long as we remember that
1
ln
(
10
)
is just a constant multipler.
Doing the problem this way gives a result of
y
'
=
1
ln
(
10
)
⋅
1
x
.