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Hyperbolic Functions
Related Topics
Wize University Calculus 1 Textbook > Derivatives
Hyperbolic Functions and their Derivatives
3 Activities
Differentiate
cosh
ln
x
\cosh \ln x
cosh
ln
x
in two ways: by simplifying before taking the derivative, and by using the derivative of
cosh
x
\cosh x
cosh
x
along with the chain rule.
1
2
−
1
2
x
2
\dfrac{1}{2}-\dfrac{1}{2x^2}
2
1
−
2
x
2
1
1
2
+
1
2
x
2
\dfrac{1}{2}+\dfrac{1}{2x^2}
2
1
+
2
x
2
1
1
3
−
1
3
x
3
\dfrac{1}{3}-\dfrac{1}{3x^3}
3
1
−
3
x
3
1
1
3
+
1
3
x
3
\dfrac{1}{3}+\dfrac{1}{3x^3}
3
1
+
3
x
3
1
None of the above
I don't know
Check Submission
More Hyperbolic Functions and their Derivatives Questions:
Derivatives: Hyperbolic Functions
Find the derivative of
f
(
x
)
=
ln
(
sinh
(
2
x
)
)
f(x) = \ln(\sinh(2x))
f
(
x
)
=
ln
(
sinh
(
2
x
))
Derivatives: Hyperbolic Functions
Q.
Find the derivative of
f
(
x
)
=
ln
(
sinh
(
2
x
)
)
f(x) = \ln(\sinh(2x))
f
(
x
)
=
ln
(
sinh
(
2
x
))
Find the derivative of
f
(
x
)
=
sinh
(
x
2
)
cosh
(
x
)
f(x) = \sinh(x^2)\cosh(x)
f
(
x
)
=
sinh
(
x
2
)
cosh
(
x
)
Hyperbolic Functions and Derivatives
Differentiate
y
=
ln
(
cosh
x
)
+
1
2
cosh
2
x
y=\ln(\cosh x)+\frac{1}{2\cosh^2 x}
y
=
ln
(
cosh
x
)
+
2
c
o
s
h
2
x
1
and simplify fully.
Example: Solve Cosh Equation
Q
:
\bf{Q:}
Q
:
Solve for
x
x
x
in the equation
cosh
x
=
13
5
\cosh x=\dfrac{13}{5}
cosh
x
=
5
13
assuming
x
>
0
x>0
x
>
0
.
Derivatives: Hyperbolic Functions
Find the derivative of
f
(
x
)
=
ln
(
sinh
(
2
x
)
)
f(x) = \ln(\sinh(2x))
f
(
x
)
=
ln
(
sinh
(
2
x
))
Find the derivative of
f
(
x
)
=
sinh
(
x
2
)
cosh
(
x
)
f(x) = \sinh(x^2)\cosh(x)
f
(
x
)
=
sinh
(
x
2
)
cosh
(
x
)
Example: Solve Cosh Equation
Q
:
\bf{Q:}
Q
:
Solve for
x
x
x
in the equation
cosh
x
=
13
5
\cosh x=\dfrac{13}{5}
cosh
x
=
5
13
assuming
x
>
0
x>0
x
>
0
.
Hyperbolic Functions and Derivatives
Differentiate
y
=
ln
(
cosh
x
)
+
1
2
cosh
2
x
y=\ln(\cosh x)+\dfrac{1}{2\cosh^2x}
y
=
ln
(
cosh
x
)
+
2
cosh
2
x
1
and simplify fully.
Find the derivative of
f
(
x
)
=
sinh
(
x
2
)
cosh
(
x
)
f(x) = \sinh(x^2)\cosh(x)
f
(
x
)
=
sinh
(
x
2
)
cosh
(
x
)
Hyperbolic Functions
Prove the following identity:
cosh
2
x
−
sinh
2
x
=
1
\cosh^2x-\sinh^2x=1
cosh
2
x
−
sinh
2
x
=
1
Find the derivative of
f
(
x
)
=
sinh
(
x
2
)
cosh
(
x
)
f(x) = \sinh(x^2)\cosh(x)
f
(
x
)
=
sinh
(
x
2
)
cosh
(
x
)
Find the derivative of
f
(
x
)
=
ln
(
sinh
(
2
x
)
)
f(x) = \ln(\sinh(2x))
f
(
x
)
=
ln
(
sinh
(
2
x
))