Evaluate d/dx (1+tan x/1-ran x )
Answers
Answer:
1
+
tan
(
x
)
1
−
tan
(
x
)
Differentiate using the Quotient Rule which states that
d
d
x
[
f
(
x
)
g
(
x
)
]
is
g
(
x
)
d
d
x
[
f
(
x
)
]
−
f
(
x
)
d
d
x
[
g
(
x
)
]
g
(
x
)
2
where
f
(
x
)
=
1
+
tan
(
x
)
and
g
(
x
)
=
1
−
tan
(
x
)
.
(
1
−
tan
(
x
)
)
d
d
x
[
1
+
tan
(
x
)
]
−
(
1
+
tan
(
x
)
)
d
d
x
[
1
−
tan
(
x
)
]
(
1
−
tan
(
x
)
)
2
Differentiate.
Tap for fewer steps...
By the Sum Rule, the derivative of
1
+
tan
(
x
)
with respect to
x
is
d
d
x
[
1
]
+
d
d
x
[
tan
(
x
)
]
.
(
1
−
tan
(
x
)
)
(
d
d
x
[
1
]
+
d
d
x
[
tan
(
x
)
]
)
−
(
1
+
tan
(
x
)
)
d
d
x
[
1
−
tan
(
x
)
]
(
1
−
tan
(
x
)
)
2
Since
1
is constant with respect to
x
, the derivative of
1
with respect to
x
is
0
.
(
1
−
tan
(
x
)
)
(
0
+
d
d
x
[
tan
(
x
)
]
)
−
(
1
+
tan
(
x
)
)
d
d
x
[
1
−
tan
(
x
)
]
(
1
−
tan
(
x
)
)
2
Add
0
and
d
d
x
[
tan
(
x
)
]
.
(
1
−
tan
(
x
)
)
d
d
x
[
tan
(
x
)
]
−
(
1
+
tan
(
x
)
)
d
d
x
[
1
−
tan
(
x
)
]
(
1
−
tan
(
x
)
)
2
The derivative of
tan
(
x
)
with respect to
x
is
sec
2
(
x
)
.
(
1
−
tan
(
x
)
)
sec
2
(
x
)
−
(
1
+
tan
(
x
)
)
d
d
x
[
1
−
tan
(
x
)
]
(
1
−
tan
(
x
)
)
2
Differentiate.
Tap for fewer steps...
By the Sum Rule, the derivative of
1
−
tan
(
x
)
with respect to
x
is
d
d
x
[
1
]
+
d
d
x
[
−
tan
(
x
)
]
.
(
1
−
tan
(
x
)
)
sec
2
(
x
)
−
(
1
+
tan
(
x
)
)
(
d
d
x
[
1
]
+
d
d
x
[
−
tan
(
x
)
]
)
(
1
−
tan
(
x
)
)
2
Since
1
is constant with respect to
x
, the derivative of
1
with respect to
x
is
0
.
(
1
−
tan
(
x
)
)
sec
2
(
x
)
−
(
1
+
tan
(
x
)
)
(
0
+
d
d
x
[
−
tan
(
x
)
]
)
(
1
−
tan
(
x
)
)
2
Add
0
and
d
d
x
[
−
tan
(
x
)
]
.
(
1
−
tan
(
x
)
)
sec
2
(
x
)
−
(
1
+
tan
(
x
)
)
d
d
x
[
−
tan
(
x
)
]
(
1
−
tan
(
x
)
)
2
Since
−
1
is constant with respect to
x
, the derivative of
−
tan
(
x
)
with respect to
x
is
−
d
d
x
[
tan
(
x
)
]
.
(
1
−
tan
(
x
)
)
sec
2
(
x
)
−
(
1
+
tan
(
x
)
)
(
−
d
d
x
[
tan
(
x
)
]
)
(
1
−
tan
(
x
)
)
2
Multiply.
Tap for more steps...
(
1
−
tan
(
x
)
)
sec
2
(
x
)
+
(
1
+
tan
(
x
)
)
d
d
x
[
tan
(
x
)
]
(
1
−
tan
(
x
)
)
2
The derivative of
tan
(
x
)
with respect to
x
is
sec
2
(
x
)
.
(
1
−
tan
(
x
)
)
sec
2
(
x
)
+
(
1
+
tan
(
x
)
)
sec
2
(
x
)
(
1
−
tan
(
x
)
)
2
Simplify.
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Apply the distributive property.
1
sec
2
(
x
)
−
tan
(
x
)
sec
2
(
x
)
+
(
1
+
tan
(
x
)
)
sec
2
(
x
)
(
1
−
tan
(
x
)
)
2
Apply the distributive property.
1
sec
2
(
x
)
−
tan
(
x
)
sec
2
(
x
)
+
1
sec
2
(
x
)
+
tan
(
x
)
sec
2
(
x
)
(
1
−
tan
(
x
)
)
2
Simplify the numerator.
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2
sec
2
(
x
)
(
1
−
tan
(
x
)
)
2
Answer:
Explanation:
1st step tan change into sin and cosx term then differentiate
Formula d(u/v)/dx=(du/dx*v-dv/dx*u)/vsquare