Gilbert Strang; Edwin “Jed” Herman

Learning Objectives

2.9.1 Apply the formulas for derivatives and integrals of the hyperbolic functions.
2.9.2 Apply the formulas for the derivatives of the inverse hyperbolic functions and their associated integrals.
2.9.3 Describe the common applied conditions of a catenary curve.

We were introduced to hyperbolic functions in Introduction to Functions and Graphs, along with some of their basic properties. In this section, we look at differentiation and integration formulas for the hyperbolic functions and their inverses.

Derivatives and Integrals of the Hyperbolic Functions

Recall that the hyperbolic sine and hyperbolic cosine are defined as

sinh x = \frac{e^{x} - e^{− x}}{2} and cosh x = \frac{e^{x} + e^{− x}}{2} .

The other hyperbolic functions are then defined in terms of $sinh x$ and $cosh x .$ The graphs of the hyperbolic functions are shown in the following figure.

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 2.81 Graphs of the hyperbolic functions.

It is easy to develop differentiation formulas for the hyperbolic functions. For example, looking at $sinh x$ we have

\begin{matrix} \frac{d}{d x} (sinh x) & = \frac{d}{d x} (\frac{e^{x} - e^{− x}}{2}) \\ = \frac{1}{2} [\frac{d}{d x} (e^{x}) - \frac{d}{d x} (e^{− x})] \\ = \frac{1}{2} [e^{x} + e^{− x}] = cosh x . \end{matrix}

Similarly, $(d / d x) cosh x = sinh x .$ We summarize the differentiation formulas for the hyperbolic functions in the following table.

$f (x)$	$\frac{d}{d x} f (x)$
$sinh x$	$cosh x$
$cosh x$	$sinh x$
$tanh x$	${sech}^{2} x$
$coth x$	$− {csch}^{2} x$
$sech x$	$− sech x tanh x$
$csch x$	$− csch x coth x$

Table 2.2 Derivatives of the Hyperbolic Functions

Let’s take a moment to compare the derivatives of the hyperbolic functions with the derivatives of the standard trigonometric functions. There are a lot of similarities, but differences as well. For example, the derivatives of the sine functions match: $(d / d x) sin x = cos x$ and $(d / d x) sinh x = cosh x .$ The derivatives of the cosine functions, however, differ in sign: $(d / d x) cos x = − sin x,$ but $(d / d x) cosh x = sinh x .$ As we continue our examination of the hyperbolic functions, we must be mindful of their similarities and differences to the standard trigonometric functions.

These differentiation formulas for the hyperbolic functions lead directly to the following integral formulas.

\begin{matrix} \int sinh u d u & = & cosh u + C & \int {csch}^{2} u d u & = & − coth u + C \\ \int cosh u d u & = & sinh u + C & \int sech u tanh u d u & = & − sech u + C \\ \int {sech}^{2} u d u & = & tanh u + C & \int csch u coth u d u & = & − csch u + C \end{matrix}

Example 2.47

Differentiating Hyperbolic Functions

Evaluate the following derivatives:

$\frac{d}{d x} (sinh (x^{2}))$
$\frac{d}{d x} {(cosh x)}^{2}$

Solution

Using the formulas in Table 2.2 and the chain rule, we get

$\frac{d}{d x} (sinh (x^{2})) = cosh (x^{2}) \cdot 2 x$
$\frac{d}{d x} {(cosh x)}^{2} = 2 cosh x sinh x$

Checkpoint 2.47

Evaluate the following derivatives:

$\frac{d}{d x} (tanh (x^{2} + 3 x))$
$\frac{d}{d x} (\frac{1}{{(sinh x)}^{2}})$

Example 2.48

Integrals Involving Hyperbolic Functions

Evaluate the following integrals:

$\int x cosh (x^{2}) d x$
$\int tanh x d x$

Solution

We can use u-substitution in both cases.

Let $u = x^{2} .$ Then, $d u = 2 x d x$ and

$\int x cosh (x^{2}) d x = \int \frac{1}{2} cosh u d u = \frac{1}{2} sinh u + C = \frac{1}{2} sinh (x^{2}) + C .$
Let $u = cosh x .$ Then, $d u = sinh x d x$ and

$\int tanh x d x = \int \frac{sinh x}{cosh x} d x = \int \frac{1}{u} d u = ln | u | + C = ln | cosh x | + C .$

Note that $cosh x > 0$ for all $x,$ so we can eliminate the absolute value signs and obtain

$\int tanh x d x = ln (cosh x) + C .$

Checkpoint 2.48

Evaluate the following integrals:

$\int {sinh}^{3} x cosh x d x$
$\int {sech}^{2} (3 x) d x$

Calculus of Inverse Hyperbolic Functions

Looking at the graphs of the hyperbolic functions, we see that with appropriate range restrictions, they all have inverses. Most of the necessary range restrictions can be discerned by close examination of the graphs. The domains and ranges of the inverse hyperbolic functions are summarized in the following table.

Function	Domain	Range
${sinh}^{−1} x$	$(− \infty, \infty)$	$(− \infty, \infty)$
${cosh}^{−1} x$	$[1, \infty)$	$[0, \infty)$
${tanh}^{−1} x$	$(−1, 1)$	$(− \infty, \infty)$
${coth}^{−1} x$	$(− \infty, −1) \cup (1, \infty)$	$(− \infty, 0) \cup (0, \infty)$
${sech}^{−1} x$	$(0, 1]$	$[0, \infty)$
${csch}^{−1} x$	$(− \infty, 0) \cup (0, \infty)$	$(− \infty, 0) \cup (0, \infty)$

Table 2.3 Domains and Ranges of the Inverse Hyperbolic Functions

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 2.82 Graphs of the inverse hyperbolic functions.

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

\begin{matrix} y & = & {sinh}^{−1} x \\ sinh y & = & x \\ \frac{d}{d x} sinh y & = & \frac{d}{d x} x \\ cosh y \frac{d y}{d x} & = & 1 . \end{matrix}

Recall that ${cosh}^{2} y - {sinh}^{2} y = 1,$ so $cosh y = \sqrt{1 + {sinh}^{2} y} .$ Then,

\frac{d y}{d x} = \frac{1}{cosh y} = \frac{1}{\sqrt{1 + {sinh}^{2} y}} = \frac{1}{\sqrt{1 + x^{2}}} .

We can derive differentiation formulas for the other inverse hyperbolic functions in a similar fashion. These differentiation formulas are summarized in the following table.

$f (x)$	$\frac{d}{d x} f (x)$
${sinh}^{−1} x$	$\frac{1}{\sqrt{1 + x^{2}}}$
${cosh}^{−1} x$	$\frac{1}{\sqrt{x^{2} - 1}}$
${tanh}^{−1} x$	$\frac{1}{1 - x^{2}}$
${coth}^{−1} x$	$\frac{1}{1 - x^{2}}$
${sech}^{−1} x$	$\frac{−1}{x \sqrt{1 - x^{2}}}$
${csch}^{−1} x$	$\frac{−1}{\| x \| \sqrt{1 + x^{2}}}$

Table 2.4 Derivatives of the Inverse Hyperbolic Functions

Note that the derivatives of ${tanh}^{−1} x$ and ${coth}^{−1} x$ are the same. Thus, when we integrate $1 / (1 - x^{2}),$ we need to select the proper antiderivative based on the domain of the functions and the values of $x .$ Integration formulas involving the inverse hyperbolic functions are summarized as follows.

\begin{matrix} \int \frac{1}{\sqrt{1 + u^{2}}} d u & = & {sinh}^{−1} u + C & \int \frac{1}{u \sqrt{1 - u^{2}}} d u & = & − {sech}^{−1} | u | + C \\ \int \frac{1}{\sqrt{u^{2} - 1}} d u & = & {cosh}^{−1} u + C & \int \frac{1}{u \sqrt{1 + u^{2}}} d u & = & − {csch}^{−1} | u | + C \\ \int \frac{1}{1 - u^{2}} d u & = & {\begin{matrix} {tanh}^{−1} u + C if | u | < 1 \\ {coth}^{−1} u + C if | u | > 1 \end{matrix} \end{matrix}

Example 2.49

Differentiating Inverse Hyperbolic Functions

Evaluate the following derivatives:

$\frac{d}{d x} ({sinh}^{−1} (\frac{x}{3}))$
$\frac{d}{d x} {({tanh}^{−1} x)}^{2}$

Solution

Using the formulas in Table 2.4 and the chain rule, we obtain the following results:

$\frac{d}{d x} ({sinh}^{−1} (\frac{x}{3})) = \frac{1}{3 \sqrt{1 + \frac{x^{2}}{9}}} = \frac{1}{\sqrt{9 + x^{2}}}$
$\frac{d}{d x} {({tanh}^{−1} x)}^{2} = \frac{2 ({tanh}^{−1} x)}{1 - x^{2}}$

Checkpoint 2.49

Evaluate the following derivatives:

$\frac{d}{d x} ({cosh}^{−1} (3 x))$
$\frac{d}{d x} {({coth}^{−1} x)}^{3}$

Example 2.50

Integrals Involving Inverse Hyperbolic Functions

Evaluate the following integrals:

$\int \frac{1}{\sqrt{4 x^{2} - 1}} d x$
$\int \frac{1}{2 x \sqrt{1 - 9 x^{2}}} d x$

Solution

We can use $u -substitution$ in both cases.

Let $u = 2 x .$ Then, $d u = 2 d x$ and we have

$\int \frac{1}{\sqrt{4 x^{2} - 1}} d x = \int \frac{1}{2 \sqrt{u^{2} - 1}} d u = \frac{1}{2} {cosh}^{−1} u + C = \frac{1}{2} {cosh}^{−1} (2 x) + C .$
Let $u = 3 x .$ Then, $d u = 3 d x$ and we obtain

$\int \frac{1}{2 x \sqrt{1 - 9 x^{2}}} d x = \frac{1}{2} \int \frac{1}{u \sqrt{1 - u^{2}}} d u = - \frac{1}{2} {sech}^{−1} | u | + C = - \frac{1}{2} {sech}^{−1} | 3 x | + C .$

Checkpoint 2.50

Evaluate the following integrals:

$\int \frac{1}{\sqrt{x^{2} - 4}} d x, x > 2$
$\int \frac{1}{\sqrt{1 - e^{2 x}}} d x$

Applications

One physical application of hyperbolic functions involves hanging cables. If a cable of uniform density is suspended between two supports without any load other than its own weight, the cable forms a curve called a catenary. High-voltage power lines, chains hanging between two posts, and strands of a spider’s web all form catenaries. The following figure shows chains hanging from a row of posts.

An image of chains hanging between posts that all take the shape of a catenary.

Figure 2.83 Chains between these posts take the shape of a catenary. (credit: modification of work by OKFoundryCompany, Flickr)

Hyperbolic functions can be used to model catenaries. Specifically, functions of the form $y = a cosh (x / a)$ are catenaries. Figure 2.84 shows the graph of $y = 2 cosh (x / 2) .$

This figure is a graph. It is of the function f(x)=2cosh(x/2). The curve decreases in the second quadrant to the y-axis. It intersects the y-axis at y=2. Then the curve becomes increasing.

Figure 2.84 A hyperbolic cosine function forms the shape of a catenary.

Example 2.51

Using a Catenary to Find the Length of a Cable

Assume a hanging cable has the shape $10 cosh (x / 10)$ for $−15 \leq x \leq 15,$ where $x$ is measured in feet. Determine the length of the cable (in feet).

Solution

Recall from Section $2.4$ that the formula for arc length is

Arc Length = \int_{a}^{b} \sqrt{1 + {[f^{'} (x)]}^{2}} d x .

We have $f (x) = 10 cosh (x / 10),$ so $f^{'} (x) = sinh (x / 10) .$ Then

\begin{matrix} Arc Length & = \int_{a}^{b} \sqrt{1 + {[f^{'} (x)]}^{2}} d x \\ = \int_{−15}^{15} \sqrt{1 + {sinh}^{2} (\frac{x}{10})} d x . \end{matrix}

Now recall that $1 + {sinh}^{2} x = {cosh}^{2} x,$ so we have

\begin{matrix} Arc Length & = \int_{−15}^{15} \sqrt{1 + {sinh}^{2} (\frac{x}{10})} d x \\ = \int_{−15}^{15} cosh (\frac{x}{10}) d x \\ = 10 sinh {(\frac{x}{10}) |}_{−15}^{15} = 10 [sinh (\frac{3}{2}) - sinh (- \frac{3}{2})] = 20 sinh (\frac{3}{2}) \\ \approx 42.586 ft . \end{matrix}

Checkpoint 2.51

Assume a hanging cable has the shape $15 cosh (x / 15)$ for $−20 \leq x \leq 20 .$ Determine the length of the cable (in feet).

Section 2.9 Exercises

377.

[T] Find expressions for $cosh x + sinh x$ and $cosh x - sinh x .$ Use a calculator to graph these functions and ensure your expression is correct.

378.

From the definitions of $cosh (x)$ and $sinh (x),$ find their antiderivatives.

379.

Show that $cosh (x)$ and $sinh (x)$ satisfy $y^{″} = y .$

380.

Use the quotient rule to verify that $tanh (x)' = {sech}^{2} (x) .$

381.

Derive ${cosh}^{2} (x) + {sinh}^{2} (x) = cosh (2 x)$ from the definition.

382.

Take the derivative of the previous expression to find an expression for $sinh (2 x) .$

383.

Prove $sinh (x + y) = sinh (x) cosh (y) + cosh (x) sinh (y)$ by changing the expression to exponentials.

384.

Take the derivative of the previous expression to find an expression for $cosh (x + y) .$

For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct.

385.

[T] $cosh (3 x + 1)$

386.

[T] $sinh (x^{2})$

387.

[T] $\frac{1}{cosh (x)}$

388.

[T] $sinh (ln (x))$

389.

[T] ${cosh}^{2} (x) + {sinh}^{2} (x)$

390.

[T] ${cosh}^{2} (x) - {sinh}^{2} (x)$

391.

[T] $tanh (\sqrt{x^{2} + 1})$

392.

[T] $\frac{1 + tanh (x)}{1 - tanh (x)}$

393.

[T] ${sinh}^{6} (x)$

394.

[T] $ln (sech (x) + tanh (x))$

For the following exercises, find the antiderivatives for the given functions.

395.

$cosh (2 x + 1)$

396.

$tanh (3 x + 2)$

397.

$x cosh (x^{2})$

398.

$3 x^{3} tanh (x^{4})$

399.

${cosh}^{2} (x) sinh (x)$

400.

${tanh}^{2} (x) {sech}^{2} (x)$

401.

$\frac{sinh (x)}{1 + cosh (x)}$

402.

$coth (x)$

403.

$cosh (x) + sinh (x)$

404.

${(cosh (x) + sinh (x))}^{n}$

For the following exercises, find the derivatives for the functions.

405.

${tanh}^{−1} (4 x)$

406.

${sinh}^{−1} (x^{2})$

407.

${sinh}^{−1} (cosh (x))$

408.

${cosh}^{−1} (x^{3})$

409.

${tanh}^{−1} (cos (x))$

410.

$e^{{sinh}^{−1} (x)}$

411.

$ln ({tanh}^{−1} (x))$

For the following exercises, find the antiderivatives for the functions.

412.

$\int \frac{d x}{4 - x^{2}}$

413.

$\int \frac{d x}{a^{2} - x^{2}}$

414.

$\int \frac{d x}{\sqrt{x^{2} + 1}}$

415.

$\int \frac{x d x}{\sqrt{x^{2} + 1}}$

416.

$\int - \frac{d x}{x \sqrt{1 - x^{2}}}$

417.

$\int \frac{e^{x}}{\sqrt{e^{2 x} - 1}}$

418.

$\int - \frac{2 x}{x^{4} - 1}$

For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation $d v / d t = g - v^{2} .$

419.

Show that $v (t) = \sqrt{g} tanh ((\sqrt{g}) t)$ satisfies this equation.

420.

Derive the previous expression for $v (t)$ by integrating $\frac{d v}{g - v^{2}} = d t .$

421.

[T] Estimate how far a body has fallen in $12$ seconds by finding the area underneath the curve of $v (t) .$

For the following exercises, use this scenario: A cable hanging under its own weight has a slope $S = d y / d x$ that satisfies $d S / d x = c \sqrt{1 + S^{2}} .$ The constant $c$ is the ratio of cable density to tension.

422.

Show that $S = sinh (c x)$ satisfies this equation.

423.

Integrate $d y / d x = sinh (c x)$ to find the cable height $y (x)$ if $y (0) = 1 / c .$

424.

Sketch the cable and determine how far down it sags at $x = 0 .$

For the following exercises, solve each problem.

425.

[T] A chain hangs from two posts $2$ m apart to form a catenary described by the equation $y = 2 cosh (x / 2) - 1 .$ Find the slope of the catenary at the left fence post.

426.

[T] A chain hangs from two posts four meters apart to form a catenary described by the equation $y = 4 cosh (x / 4) - 3 .$ Find the total length of the catenary (arc length).

427.

[T] A high-voltage power line is a catenary described by $y = 10 cosh (x / 10) .$ Find the ratio of the area under the catenary to its arc length. What do you notice?

428.

A telephone line is a catenary described by $y = a cosh (x / a) .$ Find the ratio of the area under the catenary to its arc length. Does this confirm your answer for the previous question?

429.

Prove the formula for the derivative of $y = {sinh}^{−1} (x)$ by differentiating $x = sinh (y) .$ (Hint: Use hyperbolic trigonometric identities.)

430.

Prove the formula for the derivative of $y = {cosh}^{−1} (x)$ by differentiating $x = cosh (y) .$

(Hint: Use hyperbolic trigonometric identities.)

431.

Prove the formula for the derivative of $y = {sech}^{−1} (x)$ by differentiating $x = sech (y) .$ (Hint: Use hyperbolic trigonometric identities.)

432.

Prove that ${(cosh (x) + sinh (x))}^{n} = cosh (n x) + sinh (n x) .$

433.

Prove the expression for ${sinh}^{−1} (x) .$ Multiply $x = sinh (y) = (1 / 2) (e^{y} + e^{− y})$ by $2 e^{y}$ and solve for $y .$ Does your expression match the textbook?

434.

Prove the expression for ${cosh}^{−1} (x) .$ Multiply $x = cosh (y) = (1 / 2) (e^{y} - e^{− y})$ by $2 e^{y}$ and solve for $y .$ Does your expression match the textbook?

2.9 Calculus of the Hyperbolic Functions

Derivatives and Integrals of the Hyperbolic Functions

Differentiating Hyperbolic Functions

Solution

Integrals Involving Hyperbolic Functions

Solution

Calculus of Inverse Hyperbolic Functions

Differentiating Inverse Hyperbolic Functions

Solution

Integrals Involving Inverse Hyperbolic Functions

Solution

Applications

Using a Catenary to Find the Length of a Cable

Solution

Section 2.9 Exercises