### 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