The derivative of cosh(x) with respect to x is sinh(x). One can verify this result using the definitions cosh(x) = (e^x + e^(-x))/2 and sinh(x) = (e^x - e^(-x))/2.
By definition, the derivative of e^x with respect to x is e^x. The derivative of e^(-x) with respect to x is -e^(-x). Therefore, d(cosh(x))/dx = (e^x - e^(-x))/2 = sinh(x). The graph of y = cosh(x) is a hyperbola with a local minimum at (0,1). The graph of y = sinh(x) is the slope of this hyperbola. This graph increases as x increases, and it has a point of inflection at the origin.