Hyperbolic Functions & Successive Differentiation Notes

Hyperbolic Functions & Successive Differentiation

Hyperbolic Functions
Hyperbolic functions are defined as certain combinations of ex and e -x and surprisingly have many
properties similar to trigonometric functions.



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Hyperbolic Functions Domain Range

Cosh x (-∞ , + ∞) (-∞ , + ∞)

Sinh x (-∞ , + ∞) [1, ∞)

Tanh x (-∞ , + ∞) (-1,1)

Coth x (-∞ , 0) U (0 , + ∞) (-∞ , - 1) U (1 , +∞)

Sech x (-∞ , + ∞) (0,1]

Csch x (-∞ , 0) U (0 , + ∞) (-∞ , 0) U (0 , + ∞)

, Hyperbolic Identities
1. sinh (-x) = – sinh(x)
2. cosh (-x) = cosh (x)
3. tanh (-x) = - tanh x
4. coth (-x) = - coth x
5. sech (-x) = sech x
6. csc (-x) = - csch x
7. cosh 2x = 1 + 2 sinh2(x) = 2 cosh2x - 1
8. cosh 2x = cosh2x + sinh2x
9. sinh 2x = 2 sinh x cosh x




f(x) d/dx f(x)

Sinh x Cosh x

Cosh x Sinh x

Tanh x sech²𝑥

Coth x −csch²𝑥

Sech x −sech𝑥tanh𝑥

Csch x −csch𝑥coth𝑥