Calculus of the Hyperbolic Functions: Learn It 1

  • 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:

sinhx=exex2 and coshx=ex+ex2

The other hyperbolic functions are then defined in terms of sinhx and coshx. The graphs of these functions provide insights into their behaviors.

This figure has six graphs. The first graph labeled “a” is of the function y=sinh(x). It is an increasing function from the 3rd quadrant, through the origin to the first quadrant. The second graph is labeled “b” and is of the function y=cosh(x). It decreases in the second quadrant to the intercept y=1, then becomes an increasing function. The third graph labeled “c” is of the function y=tanh(x). It is an increasing function from the third quadrant, through the origin, to the first quadrant. The fourth graph is labeled “d” and is of the function y=coth(x). It has two pieces, one in the third quadrant and one in the first quadrant with a vertical asymptote at the y-axis. The fifth graph is labeled “e” and is of the function y=sech(x). It is a curve above the x-axis, increasing in the second quadrant, to the y-axis at y=1 and then decreases. The sixth graph is labeled “f” and is of the function y=csch(x). It has two pieces, one in the third quadrant and one in the first quadrant with a vertical asymptote at the y-axis.
Figure 1. Graphs of the hyperbolic functions.

It is straightforward to develop differentiation formulas for hyperbolic functions. For instance:

ddx(sinhx)=ddx(exex2)=12[ddx(ex)ddx(ex)]=12[ex+ex]=coshx.

Similarly, (ddx)coshx=sinhx.

derivatives of the hyperbolic functions

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+Ccsch2udu=cothu+Ccoshudu=sinhu+Csechutanhudu=sechu+Csech2udu=tanhu+Ccschucothudu=cschu+C

Evaluate the following derivatives:

  1. ddx(sinh(x2))
  2. ddx(coshx)2

Evaluate the following integrals:

  1. xcosh(x2)dx
  2. tanhxdx