dy/dx if y=sqrt(e^x+1)
Answers
Step-by-step explanation:
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Calculus Examples
Popular Problems Calculus Find the Derivative - d/dx y = square root of e^x-1
y
=
√
e
x
−
1
Use
n
√
a
x
=
a
x
n
to rewrite
√
e
x
−
1
as
(
e
x
−
1
)
1
2
.
d
d
x
[
(
e
x
−
1
)
1
2
]
Differentiate using the chain rule, which states that
d
d
x
[
f
(
g
(
x
)
)
]
is
f
'
(
g
(
x
)
)
g
'
(
x
)
where
f
(
x
)
=
x
1
2
and
g
(
x
)
=
e
x
−
1
.
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1
2
(
e
x
−
1
)
1
2
−
1
d
d
x
[
e
x
−
1
]
To write
−
1
as a fraction with a common denominator, multiply by
2
2
.
1
2
(
e
x
−
1
)
1
2
−
1
⋅
2
2
d
d
x
[
e
x
−
1
]
Combine
−
1
and
2
2
.
1
2
(
e
x
−
1
)
1
2
+
−
1
⋅
2
2
d
d
x
[
e
x
−
1
]
Combine the numerators over the common denominator.
1
2
(
e
x
−
1
)
1
−
1
⋅
2
2
d
d
x
[
e
x
−
1
]
Simplify the numerator.
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1
2
(
e
x
−
1
)
−
1
2
d
d
x
[
e
x
−
1
]
Differentiate using the Sum Rule.
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1
2
(
e
x
−
1
)
1
2
(
d
d
x
[
e
x
]
+
d
d
x
[
−
1
]
)
Differentiate using the Exponential Rule which states that
d
d
x
[
a
x
]
is
a
x
ln
(
a
)
where
a
=
e
.
1
2
(
e
x
−
1
)
1
2
(
e
x
+
d
d
x
[
−
1
]
)
Differentiate using the Constant Rule.