Derivative Hyperbolic Functions

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Math2.org Math Tables: Derivatives of Hyperbolics
(Math)

Proofs of Derivatives of Hyperbolics

Proof of (d/dx)sinh(x) = cosh(x) : From the derivative of ex

Given: sinh(x) = ( ex - e-x )/2; cosh(x) = (ex + e-x)/2; (d/dx) ( f(x)+g(x) ) =(d/dx) f(x) + (d/dx) g(x); Chain Rule; (d/dx)( c*f(x) ) = c (d/dx)f(x). Solve:

(d/dx) sinh(x)= (d/dx) ( ex- e-x )/2 = 1/2 (d/dx)(ex) -1/2 (d/dx)(e-x)
= 1/2 ex + 1/2 e-x = ( ex + e-x )/2 = cosh(x)   Q.E.D

Proof of (d/dx)cosh(x) = sinh(x) : From the derivative of ex

Given: sinh(x) = ( ex - e-x )/2; cosh(x) = (ex + e-x)/2; (d/dx) ( f(x)+g(x) ) =(d/dx) f(x) +(d/dx) g(x); Chain Rule; (d/dx)( c*f(x) ) = c (d/dx)f(x). Solve:

(d/dx) cosh(x)= (d/dx) ( ex + e-x)/2 = 1/2 (d/dx)(ex) + 1/2 (d/dx)(e-x)
= 1/2 ex - 1/2 e-x = ( ex - e-x )/2 = sinh(x) Q.E.D.

Proof of (d/dx) tanh(x)= 1 - tanh2(x) : from the derivatives of sinh(x) and cosh(x)

Given: (d/dx)sinh(x) = cosh(x); (d/dx)cosh(x) = sinh(x); tanh(x) = sinh(x)/cosh(x); Quotient Rule. Solve:

(d/dx) tanh(x)= (d/dx) sinh(x)/cosh(x)
= ( cosh(x) (d/dx)sinh(x) - sinh(x) (d/dx)cosh(x) ) / cosh2(x)
= ( cosh(x) cosh(x) - sinh(x) sinh(x) ) / cosh2(x) = 1 - tanh2(x) Q.E.D.

Proof of (d/dx) csch(x)= -coth(x)csch(x), (d/dx)sech(x) = -tanh(x)sech(x), (d/dx)coth(x) = 1 - coth2(x) : From the derivatives of their reciprocal functions

Given: (d/dx)sinh(x) = cosh(x); (d/dx)cosh(x) = sinh(x); (d/dx)tanh(x) = 1 - tanh2(x); csch(x) = 1/sinh(x); sech(x) = 1/cosh(x); coth(x) = 1/tanh(x); Quotient Rule.

(d/dx) csch(x)= (d/dx) 1/sinh(x)= ( sinh(x) (d/dx)1 - 1 (d/dx) sinh(x))/sinh2(x) = -cosh(x)/sinh2(x) = -coth(x)csch(x)
(d/dx) sech(x)= (d/dx) 1/cosh(x)= ( cosh(x) (d/dx)1 - 1 (d/dx) cosh(x))/cosh2(x) = -sinh(x)/cosh2(x) = -tanh(x)sech(x)
(d/dx) coth(x)= (d/dx) 1/tanh(x)= ( tanh(x) (d/dx)1 - 1 (d/dx) tanh(x))/tanh2(x) = (tanh2(x) - 1)/tanh2(x) = 1 - coth2(x)

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