sinh2x

Sinh(2x) denotes the hyperbolic sine of the real number 2x. It is defined by the exponential identity sinh y = (e^y − e^(−y))/2, so sinh(2x) = (e^{2x} − e^{−2x})/2. Using the basic definitions of sinh and cosh, it can also be written as sinh(2x) = 2 sinh x cosh x, which is the hyperbolic double-angle formula.

As an odd function, sinh(2x) satisfies sinh(−2x) = −sinh(2x). It is differentiable on all real numbers with derivative d/dx sinh(2x) = 2 cosh(2x). The integral ∫ sinh(2x) dx equals (1/2) cosh(2x) + C. The double-angle identity also leads to various relations with the exponential forms: sinh x = (e^x − e^(−x))/2 and cosh x = (e^x + e^(−x))/2.

In applications, sinh(2x) appears in solutions to certain differential equations, in problems involving hyperbolic geometry, and in physics contexts where rapidity or Lorentz boosts are modeled. Graphically, sinh(2x) is strictly increasing for all real x, crosses the origin, and grows roughly like (1/2) e^{2x} for large positive x and −(1/2) e^{−2x} for large negative x. Its behavior reflects the general properties of the hyperbolic sine function scaled by a factor of two in its argument.