differentiation of hyperbolic functions. differentiation of hyperbolic functions by m. seppälä...
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Differentiation of Hyperbolic Functions
Differentiation of Hyperbolic Functions by M. Seppälä
coshx =ex+ e−x
2, sinhx =
ex−e−x
2,
tanhx =sinhxcoshx
, cotanh x =coshxsinhx
Hyperbolic Functions
cosh2x −sinh2 x =1
Differentiation of Hyperbolic Functions by M. Seppälä
d
dxsinhx =coshxFormulaFormula
ProofProof
d
dxsinhx =D
ex−e−x
2
⎛
⎝⎜
⎞
⎠⎟ =
D ex( ) −D e−x
( )
2
=
ex− −e−x( )
2=
ex+ e−x
2=coshx.
Differentiation of Hyperbolic Functions by M. Seppälä
d
dxcoshx =D
ex+ e−x
2
⎛
⎝⎜
⎞
⎠⎟ =
D ex( ) +D e−x
( )
2
d
dxcoshx =sinhx
=
ex+ −e−x( )
2=
ex−e−x
2=sinhx.
FormulaFormula
ProofProof
Differentiation of Hyperbolic Functions by M. Seppälä
d
dxtanh x =D
sinhxcoshx
⎛
⎝⎜⎞
⎠⎟
d
dxtanh x =
1cosh2 x
=
1cosh2 x
.
FormulaFormula
ProofProof
=
cosh2 x −sinh2 xcosh2 x
Differentiation of Hyperbolic Functions by M. Seppälä
sinhx =
ex−e−x
2
Hyperbolic Functions: Sinh
DefinitionDefinition
sinh
Differentiation of Hyperbolic Functions by M. Seppälä
sinh−1 x =ln x + x2 +1( ), x ∈°
y =sinhx ⇔ ex−e−x =2y
⇔ ex =y + y2 +1 ⇔ x =ln y + y2 +1( )
Inverse Hyperbolic Functions
FormulaFormula
ProofProof
⇔ ex
( )2−2y ex−1 =0 ⇔ ex =y ± y2 +1
Differentiation of Hyperbolic Functions by M. Seppälä
tanhx =sinhxcoshx
=ex−e−x
ex+ e−x
Hyperbolic Functions: Tanh
DefinitionDefinition
tanh
Differentiation of Hyperbolic Functions by M. Seppälä
y =tanhx ⇔ ex−e−x =y ex+ e−x
( )
tanh−1 x =
12
ln1 + x1 −x
⎛
⎝⎜⎞
⎠⎟, −1 < x <1FormulaFormula
ProofProof
⇔ 1 −y( ) ex
( )2=1 + y ⇔ ex =
1 + y1 −y
⇔ x =ln
1 + y1 −y
=12
ln1 + y1 −y
.
Differentiation of Hyperbolic Functions by M. Seppälä
sinh−1 x =ln x + x2 +1( ), x ∈°
tanh−1 x =12
ln1 + x1 −x
⎛
⎝⎜⎞
⎠⎟, −1 < x <1
Inverse Hyperbolic Functions
FormulaeFormulae
cosh−1x =ln x + x2 −1( ), x ≥1
Differentiation of Hyperbolic Functions by M. Seppälä
coshx =
ex+ e−x
2
Hyperbolic Functions: Cosh
DefinitionDefinition
cosh
0,∞⎡⎣ )
: 0 , ∞⎡⎣ ) → 1, ∞⎡
⎣ )
is an increasing bijection.
cosh−1 = cosh
0 ,∞⎡⎣ )
⎛⎝⎜
⎞⎠⎟−1
cosh−1 : 1, ∞⎡
⎣ ) → 0, ∞⎡⎣ )
Differentiation of Hyperbolic Functions by M. Seppälä
d
dxcosh−1x =
1
x2 −1, x >1 .
Derivatives of Inverse Hyperbolic Functions
FormulaFormula
Differentiation of Hyperbolic Functions by M. Seppälä
d
dxcosh−1x =
1
x2 −1, x >1FormulaFormula
ProofProof y =cosh−1 x ⇔ x =coshy, x >1, y > 0
d
dxcosh−1x =
1ddy
coshy
=
1sinhy
=1
cosh2 y −1 =
1
x2 −1
Differentiation of Hyperbolic Functions by M. Seppälä
d
dxcosh−1x =
1
x2 −1
Derivatives of Inverse Hyperbolic Functions
FormulaeFormulae
d
dxsinh−1 x =
1
1 + x2
ddx
tanh−1 x =1
1 −x2
