6.9 Hyperbolic Functions and Hanging Cables
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represent a portion of the curve x 2 − y 2 = 1, as may be seen by writing x 2 − y 2 = cosh2 t − sinh2 t = 1
and observing that x = cosh t > 0. This curve, which is shown in Figure 6.9.3b, is the right half of a larger curve called the unit hyperbola; this is the reason why the functions in this section are called hyperbolic functions. It can be shown that if t ≥ 0, then the parameter t can be interpreted as twice the shaded area in Figure 6.9.3b. (We omit the details.) DERIVATIVE AND INTEGRAL FORMULAS
Derivative formulas for sinh x and cosh x can be obtained by expressing these functions in terms of ex and e−x : d ex − e−x ex + e−x d [sinh x] = = cosh x = dx dx 2 2 d d ex + e−x ex − e−x [cosh x] = = sinh x = dx dx 2 2
Derivatives of the remaining hyperbolic functions can be obtained by expressing them in terms of sinh and cosh and applying appropriate identities. For example, d d [tanh x] = dx dx
sinh x cosh x
cosh x =
d d [sinh x] − sinh x [cosh x] dx dx cosh2 x
cosh2 x − sinh2 x 1 = = sech2 x cosh2 x cosh2 x The following theorem provides a complete list of the generalized derivative formulas and corresponding integration formulas for the hyperbolic functions. =
6.9.3
theorem d du [sinh u] = cosh u dx dx du d [cosh u] = sinh u dx dx d du [tanh u] = sech2 u dx dx du d [coth u] = −csch2 u dx dx d du [sech u] = −sech u tanh u dx dx du d [csch u] = −csch u coth u dx dx
cosh u du = sinh u + C sinh u du = cosh u + C sech2 u du = tanh u + C csch2 u du = − coth u + C sech u tanh u du = −sech u + C csch u coth u du = −csch u + C
Example 2 d d 3 [cosh(x 3 )] = sinh(x 3 ) · [x ] = 3x 2 sinh(x 3 ) dx dx d 1 d sech2 x [ln(tanh x)] = · [tanh x] = dx tanh x dx tanh x