tanhx2

tanhx2 is a mathematical expression that represents the hyperbolic tangent function applied to the square of the variable x. It is typically written as tanh(x^2). The hyperbolic tangent function, denoted as tanh(u), is defined as the ratio of the hyperbolic sine (sinh(u)) to the hyperbolic cosine (cosh(u)). Mathematically, tanh(u) = sinh(u) / cosh(u) = (e^u - e^-u) / (e^u + e^-u). Therefore, tanhx2 is equivalent to (e^(x^2) - e^(-x^2)) / (e^(x^2) + e^(-x^2)).

The domain of tanhx2 is all real numbers, as x^2 is defined for all real x, and the hyperbolic tangent function is defined for all real inputs. The range of tanhx2 is between 0 (exclusive) and 1 (exclusive). As x approaches 0, x^2 approaches 0, and tanh(0) = 0, so tanhx2 approaches 0. As the absolute value of x increases, x^2 increases without bound. The hyperbolic tangent function approaches 1 as its input approaches infinity and -1 as its input approaches negative infinity. Since x^2 is always non-negative, the input to the tanh function is always non-negative. Thus, as |x| approaches infinity, x^2 approaches infinity, and tanhx2 approaches tanh(infinity), which is 1.

The function tanhx2 is an even function, meaning that f(-x) = f(x). This is because (-x)^2 = x^2, so tanh((-x)^2) = tanh(x^2). Its graph is symmetric about the y-axis. The derivative of tanhx2 can be found using the chain rule. Let u = x^2. Then the derivative of tanhu with respect to u is sech^2(u), where sech(u) is the hyperbolic secant function. The derivative of u = x^2 with respect to x is 2x. Therefore, the derivative of tanhx2 with respect to x is sech^2(x^2) * 2x.