Coshasinhbx

Coshasinhbx is a mathematical function, specifically a hyperbolic function, that represents the hyperbolic cosine of the inverse hyperbolic sine of a variable x. It can be expressed as cosh(arsinh(x)). To understand coshasinhbx, it's helpful to first consider its components. The hyperbolic sine function, sinh(x), is defined as (e^x - e^-x) / 2. The inverse hyperbolic sine function, arsinh(x), is its inverse, meaning if y = sinh(x), then x = arsinh(y). It can also be expressed using logarithms as arsinh(x) = ln(x + sqrt(x^2 + 1)). The hyperbolic cosine function, cosh(x), is defined as (e^x + e^-x) / 2.

Therefore, coshasinhbx = cosh(arsinh(x)). Substituting the logarithmic form of arsinh(x) into the definition of cosh(x) results in a simplified expression. Let y = arsinh(x) = ln(x + sqrt(x^2 + 1)). Then cosh(y) = (e^y + e^-y) / 2. Substituting the expression for y, we get cosh(arsinh(x)) = (e^(ln(x + sqrt(x^2 + 1))) + e^(-ln(x + sqrt(x^2 + 1)))) / 2. Using the property e^(ln(a)) = a, this simplifies to (x + sqrt(x^2 + 1) + 1 / (x + sqrt(x^2 + 1))) / 2. Further algebraic manipulation can lead to an even more compact form, which is sqrt(x^2 + 1). Thus, coshasinhbx = sqrt(x^2 + 1). This function is defined for all real numbers x and its graph is a curve that resembles the upper branch of a hyperbola.