1. 3.11 Definitions of Hyperbolic Functions
sinh x =
ex − e−x
2
cosh x =
ex + e−x
2
tanh x =
sinh x
cosh x
coth x =
1
tanh x
sech x =
1
cosh x
csch x =
1
sinh x
2. Derivatives
d
sinh x = cosh x
dx
d
cosh x = sinh x
dx
d
tanh x = sech2 x
dx
d
coth x = −csch2 x
dx
d
sech x = −sech x tanh x
dx
d
csch x = −csch x coth x
dx
1
3.11 Hyperbolic Functions
2
3. Identities
sinh(−x) = − sinh x
cosh(−x) = cosh x
cosh2 x − sinh2 x = 1
1 − tanh2 x = sech2 x
sinh(x + y) = sinh x cosh y + cosh x sinh y
cosh(x + y) = cosh x cosh y + sinh x sinh y
4. Inverse Hyperbolic Functions
y = sinh−1 x
⇔ x = sinh y
y = cosh−1 x ⇔ x = cosh y
and y ≥ 0
y = tanh−1 x ⇔ x = tanh y
5. Derivatives
d
1
sinh−1 x = √
dx
1 + x2
d
1
cosh−1 x = √
2
dx
x −1
d
1
tanh−1 x =
dx
1 − x2
d
1
coth−1 x =
dx
1 − x2
d
1
sech−1 x = − √
dx
x 1 − x2
d
1
csch−1 x = − √
dx
|x| x2 + 1
3.11 Hyperbolic Functions
3
6. Examples
Example 6.1. Find the numeric value of sinh 1 and sinh−1 1.
Example 6.2. Given sech x = 2/7, find the other hyperbolic functions.
Example 6.3. Find lim coth x
x→−∞
Example 6.4. Find lim+ coth x
x→0
Example 6.5. Find the derivative of y = x2 cosh(3x − 1)
√
Example 6.6. Find the derivative of y = sech−1 1 − x2
Example 6.7. Find the value(s) for x where the slope of the tangent to the graph of
y = tanh−1 x is 2.