- Differentiate and integrate hyperbolic functions and their inverse forms
- Understand the practical situations where the catenary curve appears
Derivatives and Integrals of the Hyperbolic Functions
The hyperbolic sine (sinh) and hyperbolic cosine (cosh) functions are defined as:
The other hyperbolic functions are then defined in terms of sinhx and coshx. The graphs of these functions provide insights into their behaviors.

It is straightforward to develop differentiation formulas for hyperbolic functions. For instance:
Similarly, (ddx)coshx=sinhx.
derivatives of the hyperbolic functions
f(x) | ddxf(x) |
---|---|
sinhx | coshx |
coshx | sinhx |
tanhx | sech2x |
cothx | −csch2x |
sechx | −sechxtanhx |
cschx | −cschxcothx |
The derivatives of hyperbolic functions share similarities with those of trigonometric functions. For example:
(ddx)sinx=cosx and (ddx)sinhx=coshx.
However, the derivatives of the cosine functions differ in sign:
(ddx)cosx=−sinx, but (ddx)coshx=sinhx.
Using the differentiation formulas, we can derive the integral formulas for hyperbolic functions.
integral formulas for hyperbolic functions
∫sinhudu=coshu+C∫csch2udu=−cothu+C∫coshudu=sinhu+C∫sechutanhudu=−sechu+C∫sech2udu=tanhu+C∫cschucothudu=−cschu+C
Evaluate the following derivatives:
- ddx(sinh(x2))
- ddx(coshx)2
Evaluate the following integrals:
- ∫xcosh(x2)dx
- ∫tanhxdx