The hyperbolic functions are defined in terms of certain combinations of ex and e−x. These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes. Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary.
The shape of a strand of silk in a spider’s web can be described in terms of a hyperbolic function. The same shape applies to a chain or cable hanging from two supports with only its own weight.
The hyperbolic sine and hyperbolic cosine are defined as
sinhx=ex−e−x2 and coshx=ex+e−x2
The other hyperbolic functions are then defined in terms of sinhx and coshx.
Evaluate the following derivative.
Enclose numerators and denominators in parentheses. For example, a − b / 1 + n . Include a multiplication sign between symbols and enclose arguments of functions in parentheses. For example, tanh a * x .
ddxtanhx6+7x=
ddx1sinhx5=
Find the derivatives of fx=tanhx2+1.
Include a multiplication sign between symbols. For example, a * x . Enclose arguments of functions, numerators, and denominators in parentheses. For example, sina*x or a − b / 1 + n .
d dx tanh x 2 + 1 =