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 Proofs: Derivatives Hyperbolas (Math | Calculus | Derivatives | Table Of | Hyperbolas)
 sinh(x) = cosh(x) cosh(x) = sinh(x) tanh(x) = 1 - tanh(x)^2 csch(x) = -coth(x)csch(x) sech(x) = -tanh(x)sech(x) coth(x) = 1 - coth(x)^2

## Proofs of Derivatives of Hyperbolas

Proof of sinh(x) = cosh(x) : From the derivative of e^x

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

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

Proof of cosh(x) = sinh(x) : From the derivative of e^x

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

cosh(x)= ( e^x + e^-x)/2 = 1/2 (e^x) + 1/2 (e^-x)
= 1/2 e^x - 1/2 e^-x = ( e^x - e^-x )/2 = sinh(x)       QED

Proof of tanh(x)= 1 - tan^2(x) : from the derivatives of sinh(x) and cosh(x)

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

tanh(x)= sinh(x)/cosh(x)
= ( cosh(x) sinh(x) - sinh(x) cosh(x) ) / cosh^2(x)
= ( cosh(x) cosh(x) - sinh(x) sinh(x) ) / cosh^2(x) = 1 - tanh^2(x)       QED

Proof of csch(x)= -coth(x)csch(x), sech(x) = -tanh(x)sech(x), coth(x) = 1 - coth^2(x) : From the derivatives of their reciprocal functions

csch(x)= 1/sinh(x)= ( sinh(x) 1 - 1 sinh(x))/sinh^2(x) = -cosh(x)/sinh^2(x) = -coth(x)csch(x)
sech(x)= 1/cosh(x)= ( cosh(x) 1 - 1 cosh(x))/cosh^2(x) = -sinh(x)/cosh^2(x) = -tanh(x)sech(x)
coth(x)= 1/tanh(x)= ( tanh(x) 1 - 1 tanh(x))/tanh^2(x) = (tanh^2(x) - 1)/tanh^2(x) = 1 - coth^2(x)