Calculus of the Hyperbolic Functions: Learn It 2

Calculus of Inverse Hyperbolic Functions

Looking at the graphs of the hyperbolic functions, we see that with appropriate range restrictions, they all have inverses. The inverse hyperbolic functions have specific domain and range restrictions, summarized in the table below:

Domains and Ranges of the Inverse Hyperbolic Functions
Function Domain Range
sinh1x (,) (,)
cosh1x (1,) [0,)
tanh1x (1,1) (,)
coth1x (,1)(1,) (,0)(0,)
sech1x (0, 1) [0,)
csch1x (,0)(0,) (,0)(0,)

The graphs of the inverse hyperbolic functions are shown in the following figure.

This figure has six graphs. The first graph labeled “a” is of the function y=sinh^-1(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^-1(x). It is in the first quadrant, beginning on the x-axis at 2 and increasing. The third graph labeled “c” is of the function y=tanh^-1(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^-1(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^-1(x). It is a curve decreasing in the first quadrant and stopping on the x-axis at x=1. The sixth graph is labeled “f” and is of the function y=csch^-1(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 2. Graphs of the inverse hyperbolic functions.

To find the derivatives of the inverse functions, we use implicit differentiation. We have

y=sinh1xsinhy=xddxsinhy=ddxxcoshydydx=1.

Recall that cosh2ysinh2y=1, so coshy=1+sinh2y. Then,

dydx=1coshy=11+sinh2y=11+x2.

We can derive differentiation formulas for the other inverse hyperbolic functions in a similar fashion. 

derivatives of inverse hyperbolic functions

Derivatives of the Inverse Hyperbolic Functions
f(x) ddxf(x)
sinh1x 11+x2
cosh1x 1x21
tanh1x 11x2
coth1x 11x2
sech1x 1x1x2
csch1x 1|x|1+x2

Note that the derivatives of tanh1x and coth1x are the same. Thus, when we integrate 1/(1x2), we need to select the proper antiderivative based on the domain of the functions and the values of x.

 integral formulas for inverse hyperbolic functions

11+u2du=sinh1u+C1u1u2du=sech1|u|+C1u21du=cosh1u+C1u1+u2du=csch1|u|+C11u2du={tanh1u+C if |u|<1coth1u+C if |u|>1

Evaluate the following derivatives:

  1. ddx(sinh1(x3))
  2. ddx(tanh1x)2

Evaluate the following integrals:

  1. 1x24dx,x>2
  2. 11e2xdx