Answer Happy

1. The hyperbolic functions cosh and sinh are defined by the formulas pit e-x cosh(x) = eª + ex 9 2 sinh(2) = 2 The functions tanh, coth, sech and csch are defined in terms of cosh and sinh analogously to how they are for trigonometric functions: tanh() sinh(2) cosh(x)' coth(x) cosh(x) sinh() 1 1 sech(x): csch(x) cosh(x) sinh(r) (a) Find formulas for the derivatives of all six of these functions. You must show all of your work. = =

(b) The function sinh is one-to-one on R, and its range is R, so it has an inverse defined on R, which we call arcsinh. Use implicit differentiation to prove that 1 (arcsinh(r)) = = x² + 1