csch2

csch2 is a hyperbolic trigonometric function. It is defined as the hyperbolic cosecant of twice the argument, or csch(2x). The hyperbolic cosecant function, csch(x), is the reciprocal of the hyperbolic sine function, sinh(x). Therefore, csch2(x) can be expressed as 1/sinh(2x).

The hyperbolic sine function is defined in terms of the exponential function as sinh(x) = (e^x - e^(-x))/2. Using this definition, csch2(x) can be written as 2/(e^(2x) - e^(-2x)).

The domain of csch2(x) is all real numbers except for values where the denominator is zero. The denominator, sinh(2x), is zero when 2x = 0, which means x = 0. Thus, the domain of csch2(x) is all real numbers except x = 0.

The range of csch2(x) is all real numbers except for the interval (-2, 2).

The derivative of csch2(x) is -2 csch2(2x) coth(2x). The integral of csch2(x) is (1/2) coth(x) + C, where C is the constant of integration.

csch2(x) is an odd function, meaning that csch2(-x) = -csch2(x). It has a vertical asymptote at x = 0.