# Hyperbolic Formulas

Posted By on December 21, 2014

### Definitions of hyperbolic functions

 sinhx=ex−e−x2
 coshx=ex+e−x2
 tanhx=ex−e−xex+e−x=sinhxcoshx
 cschx=2ex−e−x=1sinhx
 sechx=2ex+e−x=1coshx
 cothx=ex+e−xex−e−x=coshxsinhx

### Derivatives

 ddxsinhx=coshx
 ddxcoshx=sinhx
 ddxtanhx=sech2x
 ddxcschx=−cschx⋅cothx
 ddxsechx=−sechx⋅tanhx
 ddxcothx=−csch2x

### Hyperbolic identities

 cosh2x−sinh2x=1
 tanh2x+sech2x=1
 coth2x−csch2x=1
 sinh(x±y)=sinhx⋅coshy±coshx⋅sinhy
 cosh(x±y)=coshx⋅coshy±sinhx⋅sinhy
 sinh(2⋅x)=2⋅sinhx⋅coshx
 cosh(2⋅x)=cosh2x+sinh2x
 sinh2x=−1+cosh2x2
 cosh2x=1+cosh2x2

### Inverse Hyperbolic functions

 sinh−1x=ln(x+x2+1−−−−−√),  x∈(−∞,∞)
 cosh−1x=ln(x+x2−1−−−−−√),  x∈[1,∞)
 tanh−1x=12ln(1+x1−x),  x∈(−1,1)
 coth−1x=12ln(x+1x−1),  x∈(−∞,−1)∪(1,∞)
 sech−1x=ln(1+1−x2−−−−−√x),  x∈(0,1]
 csch−1x=ln(1x+1−x2−−−−−√|x|),  x∈(−∞,0)∪(0,∞)

### Derivatives of Inverse Hyperbolic functions

 ddxsinh−1x=1x2+1−−−−−√
 ddxcosh−1x=1x2−1−−−−−√
 ddxtanh−1x=11−x2
 ddxcsch−1x=−1|x|1+x2−−−−−√
 ddxsech−1x=−1x1−x2−−−−−√
 ddxcoth−1x=11−x2

#### Posted by Akash Kurup

Founder and C.E.O, World4Engineers Educationist and Entrepreneur by passion. Orator and blogger by hobby

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