Derivative of hyperbolic functions.

df/dx = = {[dg/dx]h - g[dh/dx]}/h2 when h≠0 and f = g/h
f, g and h are functions
We know that
cosh x = (ex + e-x)/2
sinh x = (ex - e-x)/2
tanh x = sinh x/cosh x
coth x = cosh x/sinh x
sech x = 1/cosh x
cosech x = 1/sinh x



Derivative of cosh x
d(cosh x)/dx
= d[(ex + e-x)/2]/dx
= d(ex/2)/dx + d(e-x/2)/dx
= ex/2 - (e-x)/2
= (ex - e-x)/2
= sinh x


Derivative of sinh x
d(sinh x)/dx
= d[(ex - e-x)/2]/dx
= d(ex/2)/dx - d(e-x/2)/dx
= ex/2 + (e-x)/2
= (ex + e-x)/2
= cosh x


Derivative of tanh x

tanh x = sinh x /cosh x

d(tanh x)/dx
= d(sinh x/cosh x)/dx
= [d(sinh x)/dx cosh x - sinh x d(cosh x)/dx]/cosh2 x
= [cosh x cosh x - sinh x sinh x)]/cosh2 x
= [cosh2 x - sinh2 x]/cosh2 x
= 1/cosh2 x = sech2 x


Derivative of coth x

coth x = cosh x /sinh x

d(coth x)/dx
= d(cosh x/sinh x)/dx
= [d(cosh x)/dx sinh x - cosh x d(sinh x)/dx]/sinh2 x
= [sinh x sinh x - cosh x cosh x ]/sinh2 x
= -[cosh2 x - sinh2 x]/sinh2 x
= -1/sinh2 x = -cosech2 x


Derivative of sech x

sech x = 1/cosh x

d(sech x)/dx
= d(1/cosh x)/dx
= [d(1)/dx cosh x - 1 d(cosh x)/dx]/cosh2 x
= [ -sinh x ]/cosh2 x
= -sech x tanh x


Derivative of cosech x

cosech x = 1/sinh x

d(cosech x)/dx
= d(1/sinh x)/dx
= [d(1)/dx sinh x - 1 d(sinh x)/dx]/sinh2 x
= [ -cosh x ]/sinh2 x
= -cosech x coth x





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