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


Proofs of Derivatives of Hyperbolas

Proof of (d/dx)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; (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) ( e^x- e^-x )/2 = 1/2 (d/dx)(e^x) -1/2 (d/dx)(e^-x)
= 1/2 e^x + 1/2 e^-x = ( e^x + e^-x )/2 = cosh(x)       Q.E.D

Proof of (d/dx)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; (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) ( e^x + e^-x)/2 = 1/2 (d/dx)(e^x) + 1/2 (d/dx)(e^-x)
= 1/2 e^x - 1/2 e^-x = ( e^x - e^-x )/2 = sinh(x)       QED

Proof of (d/dx) tanh(x)= 1 - tan^2(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) ) / cosh^2(x)
= ( cosh(x) cosh(x) - sinh(x) sinh(x) ) / cosh^2(x) = 1 - tanh^2(x)       QED

Proof of (d/dx) csch(x)= -coth(x)csch(x), (d/dx)sech(x) = -tanh(x)sech(x), (d/dx)coth(x) = 1 - coth^2(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 - tanh^2(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))/sinh^2(x) = -cosh(x)/sinh^2(x) = -coth(x)csch(x)
(d/dx) sech(x)= (d/dx) 1/cosh(x)= ( cosh(x) (d/dx)1 - 1 (d/dx) cosh(x))/cosh^2(x) = -sinh(x)/cosh^2(x) = -tanh(x)sech(x)
(d/dx) coth(x)= (d/dx) 1/tanh(x)= ( tanh(x) (d/dx)1 - 1 (d/dx) tanh(x))/tanh^2(x) = (tanh^2(x) - 1)/tanh^2(x) = 1 - coth^2(x)

  
 
  

 
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