How to derive the formulas of hyperbolic functions?

The hyperbolic functions are derived from the exponential function e^x.

To derive the hyperbolic functions, we start with the exponential function e^x and its inverse, the natural logarithm ln(x). We define the hyperbolic functions as follows:

sinh(x) = (e^x - e^-x)/2
cosh(x) = (e^x + e^-x)/2
tanh(x) = sinh(x)/cosh(x)
sech(x) = 1/cosh(x)
csch(x) = 1/sinh(x)
coth(x) = cosh(x)/sinh(x)

We can derive these formulas by using the definitions of the exponential function and the natural logarithm. For example, to derive the formula for sinh(x), we start with:

sinh(x) = (e^x - e^-x)/2

We can rewrite this using the definition of the exponential function:

sinh(x) = (1/2)(e^x - 1/e^x)

We can then use the identity e^x + 1/e^x = 2cosh(x) to get:

sinh(x) = (1/2)(e^x - 1/e^x) = (1/2)(e^x - e^-x)/2

Similarly, we can derive the formulas for cosh(x), tanh(x), sech(x), csch(x), and coth(x) using similar techniques.

Overall, the hyperbolic functions are derived from the exponential function e^x and its inverse, the natural logarithm ln(x), by defining them in terms of e^x and e^-x. These functions have many important applications in mathematics and physics, particularly in the study of curves and surfaces.