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1.)
y
=
x
tan
x
The first step is to take the natural log of both sides:
2.)
ln
y
=
ln
x
tan
x
Using the exponents property of logarithms, we bring the exponent out in front of the log as a multiplier. This is done to make differentiating easier:
3.)
ln
y
=
tan
x
⋅
ln
x
Now we implicitly differentiate, taking care to use the chain rule on
ln
y
. We will also apply the product rule to the right side of the equation:
4.)
1
y
⋅
d
y
d
x
=
d
d
x
[
tan
x
]
⋅
ln
x
+
d
d
x
[
ln
x
]
⋅
tan
x
We know that the derivative of
tan
x
is equal to
sec
2
x
, and the derivative of
ln
x
is
1
x
:
5.)
1
y
⋅
d
y
d
x
=
sec
2
x
ln
x
+
tan
x
x
Multiply both sides by
y
to isolate
d
y
d
x
:
6.)
d
y
d
x
=
y
(
sec
2
x
ln
x
+
tan
x
x
)
We know
y
from step 1, so we will substitute:
7.)
d
y
d
x
=
x
tan
x
(
sec
2
x
ln
x
+
tan
x
x
)
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