Find the derivative of = /
^2
Answers
Answer:
The idea here is that you can use the fact that you know what the derivative of
e
x
is to try and determine what the derivative of another constant raised to the power of
x
, in this case equal to
2
, is.
To do that, you need to write
2
as an exponential number that has the base equal to
e
.
Use the fact that
e
ln
(
a
)
=
a
to write
e
ln
2
=
2
This implies that
2
x
will be equivalent to
2
x
=
(
e
ln
2
)
x
=
e
x
⋅
ln
2
Your derivative now looks like this
d
d
x
(
e
x
⋅
ln
2
)
This is where the chain rule comes into play. You know that the derivative of a function
y
=
f
(
u
)
can be written as
d
y
d
x
=
d
y
d
u
⋅
d
u
d
x
In your case,
y
=
e
x
⋅
ln
2
, and
u
=
x
⋅
ln
2
, so that your derivative becomes
d
d
x
(
e
u
)
=
e
u
d
u
=
e
u
⋅
d
d
x
(
u
)
d
d
x
(
e
u
)
=
e
u
⋅
d
d
x
(
u
)
Now replace
u
to calculate
d
d
x
(
u
)
d
d
x
(
e
x
⋅
ln
2
)
=
e
x
⋅
ln
2
⋅
d
d
x
(
x
⋅
ln
2
)
d
d
x
(
e
x
⋅
ln
2
)
=
e
x
⋅
ln
2
⋅
ln
2
d
d
x
(
x
)
d
d
x
(
e
x
⋅
ln
2
)
=
e
x
⋅
ln
2
⋅
ln
2
Therefore,
d
d
x
(
2
x
)
=
2
x
⋅
ln
2