 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
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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
Given: sinh(x)
= cosh(x); cosh(x)
= sinh(x); tanh(x)
= 1 - tanh^2(x); csch(x)
= 1/sinh(x); sech(x) = 1/cosh(x); coth(x) = 1/tanh(x); Quotient
Rule.
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)
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