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Hyperbolic Functions

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**The Hyperbolic Sine, Hyperbolic Cosine & Hyperbolic Tangent**

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**The Inverse Hyperbolic Cotangent, Hyperbolic Secant & Hyperbolic Cosecant**

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Values

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**Remember Logarithmic Identities ( Calculus I )**

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**Logarithmic Identities II**

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Identities

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Proofs Identity (1)

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Proofs Identity (2)

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Proofs Identity (3)

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Proofs Identity (4)

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Proofs Identity (5)

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Proofs Identity (6)

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Proofs Identity (7)

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**Derivatives of Hyperbolic Functions**

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**Proofs (1) (sinhx)’ = coshx (2) (coshx)’ = sinhx**

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**Proofs (3) (tanhx)’ = sech2x**

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**Proofs (4) (cothx)’ = -csch2x**

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**Proofs (5) (sechx)’ = - sechx tanhx**

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**Proofs (5) (cschx)’ = - cschx cothx**

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**Integrals Involving hyperbolic Functions**

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Examples I

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Examples II

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Examples III

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Examples IV

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Examples V

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**Graphs of Hyperbolic Functions**

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**Graphs of Exponential Functions Functions**

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**Graphs of Exponential Functions**

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f(x) = Cosh x

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f(x) = cosh x = (½)ex + (½) e-x as x → ∞ the values f(x) → ∞ following (½)ex as x → - ∞ the values f(x) → ∞ following (½) e-x

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f(x) = sinh x

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f(x) = tanh x

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f(x) = secx

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