3 Derivatives 3.9 Linear Approximations and Differentials 4 Applications of Derivatives

3.10 Derivatives of the Hyperbolic Functions

Coming Concepts

•

Identify the hyperbolic functions, their graphs, and basic identities.
•

Apply the formulas for derivatives of the hyperbolic functions.
•

Apply the formulas for the derivatives of the inverse hyperbolic functions.
•

Describe the common applied conditions of a catenary curve.

3.10.1 Hyperbolic Functions

The hyperbolic functions are defined in terms of certain combinations of $\displaystyle e^{x}$ and $\displaystyle e^{-x}$ . 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 (Figure 3.28). If we introduce a coordinate system so that the low point of the chain lies along the $\displaystyle y$ -axis, we can describe the height of the chain in terms of a hyperbolic function. First, we define the hyperbolic functions.

A photograph of a spider web collecting dew drops. — Figure 3.28: 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. (credit: “Mtpaley”, Wikimedia Commons)

Definition.

Hyperbolic cosine

\displaystyle\cosh x=\frac{e^{x}+e^{-x}}{2}

Hyperbolic sine

\displaystyle\sinh x=\frac{e^{x}-e^{-x}}{2}

Hyperbolic tangent

\displaystyle\tanh x=\frac{\sinh x}{\cosh x}=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}

Hyperbolic cosecant

\displaystyle\text{csch}x=\frac{1}{\sinh x}=\frac{2}{e^{x}-e^{-x}}

Hyperbolic secant

\displaystyle\text{sech}x=\frac{1}{\cosh x}=\frac{2}{e^{x}+e^{-x}}

Hyperbolic cotangent

\displaystyle\coth x=\frac{\cosh x}{\sinh x}=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}

The name cosh rhymes with “gosh,” whereas the name sinh is pronounced “cinch.” Tanh, sech, csch, and coth are pronounced “tanch,” “seech,” “coseech,” and “cotanch,” respectively.

Using the definition of $\displaystyle\cosh(x)$ and principles of physics, it can be shown that the height of a hanging chain, such as the one in Figure 3.28, can be described by the function $\displaystyle h(x)=a\cosh(x/a)+c$ for certain constants $\displaystyle a$ and $\displaystyle c$ .

But why are these functions called hyperbolic functions? To answer this question, consider the quantity $\displaystyle\cosh^{2}t-\sinh^{2}t$ . Using the definition of $\displaystyle\cosh$ and $\displaystyle\sinh$ , we see that

\displaystyle\cosh^{2}t-\sinh^{2}t=\frac{e^{2t}+2+e^{-2t}}{4}-\frac{e^{2t}-2+e% ^{-2t}}{4}=1.

This identity is the analog of the trigonometric identity $\displaystyle\cos^{2}t+\sin^{2}t=1$ . Here, given a value $\displaystyle t$ , the point $\displaystyle(x,y)=(\cosh t,\sinh t)$ lies on the unit hyperbola $\displaystyle x^{2}-y^{2}=1$ (Figure 3.29).

Figure 3.29: The unit hyperbola

\displaystyle\cosh^{2}t-\sinh^{2}t=1

3.10.2 Graphs of Hyperbolic Functions

To graph $\displaystyle\cosh x$ and $\displaystyle\sinh x$ , we make use of the fact that both functions approach $\displaystyle(1/2)e^{x}$ as $\displaystyle x\to\infty$ , since $\displaystyle e^{-x}\to 0$ as $\displaystyle x\to\infty$ . As $\displaystyle x\to-\infty,\cosh x$ approaches $\displaystyle 1/2e^{-x}$ , whereas $\displaystyle\sinh x$ approaches $\displaystyle-1/2e^{-x}$ . Therefore, using the graphs of $\displaystyle 1/2e^{x},1/2e^{-x}$ , and $\displaystyle-1/2e^{-x}$ as guides, we graph $\displaystyle\cosh x$ and $\displaystyle\sinh x$ . To graph $\displaystyle\tanh x$ , we use the fact that $\displaystyle\tanh(0)=0,-1<\tanh(x)<1$ for all $\displaystyle x,\tanh x\to 1$ as $\displaystyle x\to\infty$ , and $\displaystyle\tanh x\to-1$ as $\displaystyle x\to-\infty$ . The graphs of the other three hyperbolic functions can be sketched using the graphs of $\displaystyle\cosh x,\sinh x$ , and $\displaystyle\tanh x$ (Figure 3.30).

Figure 3.30: The hyperbolic functions involve combinations of

\displaystyle e^{x}

and

\displaystyle e^{-x}

3.10.3 Identities Involving Hyperbolic Functions

The identity $\displaystyle\cosh^{2}t-\sinh^{2}t$ , shown in Figure 3.29, is one of several identities involving the hyperbolic functions, some of which are listed next. The first four properties follow easily from the definitions of hyperbolic sine and hyperbolic cosine. Except for some differences in signs, most of these properties are analogous to identities for trigonometric functions.

Rule 3.10.1 (Identities Involving Hyperbolic Functions).

1.

$\displaystyle\cosh(-x)=\cosh x$
2.

$\displaystyle\sinh(-x)=-\sinh x$
3.

$\displaystyle\cosh x+\sinh x=e^{x}$
4.

$\displaystyle\cosh x-\sinh x=e^{-x}$
5.

$\displaystyle\cosh^{2}x-\sinh^{2}x=1$
6.

$\displaystyle 1-\tanh^{2}x=\text{sech}^{2}x$
7.

$\displaystyle\coth^{2}x-1=\text{csch}^{2}x$
8.

$\displaystyle\sinh(x\pm y)=\sinh x\cosh y\pm\cosh x\sinh y$
9.

$\displaystyle\cosh(x\pm y)=\cosh x\cosh y\pm\sinh x\sinh y$

Example 3.10.1 (Evaluating Hyperbolic Functions).

(a)

Simplify $\displaystyle\sinh(5\ln x)$ .
(b)

If $\displaystyle\sinh x=3/4$ , find the values of the remaining five hyperbolic functions.

Solution:

(a)

Using the definition of the $\displaystyle\sinh$ function, we write

$\displaystyle\sinh(5\ln x)=\frac{e^{5\ln x}-e^{-5\ln x}}{2}=\frac{e^{\ln(x^{5}% )}-e^{\ln(x^{-5})}}{2}=\frac{x^{5}-x^{-5}}{2}.$
(b)

Using the identity $\displaystyle\cosh^{2}x-\sinh^{2}x=1$ , we see that

$\displaystyle\cosh^{2}x=1+\left(\frac{3}{4}\right)^{2}=\frac{25}{16}.$

Since $\displaystyle\cosh x\geq 1$ for all $\displaystyle x$ , we must have $\displaystyle\cosh x=5/4$ . Then, using the definitions for the other hyperbolic functions, we conclude that $\displaystyle\tanh x=3/5,\text{csch}x=4/3,\text{sech}x=4/5$ , and $\displaystyle\coth x=5/3$ .

Checkpoint 3.10.1.

Simplify $\displaystyle\cosh(2\ln x)$ .

Hint: Use the definition of the cosh function and the power property of logarithm functions.

3.10.4 Inverse Hyperbolic Functions

From the graphs of the hyperbolic functions, we see that all of them are one-to-one except $\displaystyle\cosh x$ and $\displaystyle\text{sech}x$ . If we restrict the domains of these two functions to the interval $\displaystyle[0,\infty)$ , then all the hyperbolic functions are one-to-one, and we can define the inverse hyperbolic functions. Since the hyperbolic functions themselves involve exponential functions, the inverse hyperbolic functions involve logarithmic functions.

Definition.

Inverse Hyperbolic Functions

	$\displaystyle\sinh^{-1}x=\text{arcsinh}x=\ln\left(x+\sqrt{x^{2}+1}\right)$			$\displaystyle\cosh^{-1}x=\text{arccosh}x=\ln\left(x+\sqrt{x^{2}-1}\right)$
	$\displaystyle\tanh^{-1}x=\text{arctanh}x=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$			$\displaystyle\coth^{-1}x=\text{arccot}x=\frac{1}{2}\ln\left(\frac{x+1}{x-1}\right)$
	$\displaystyle\text{sech}^{-1}x=\text{arcsech}x=\ln\left(\frac{1+\sqrt{1-x^{2}}% }{x}\right)$			$\displaystyle\text{csch}^{-1}x=\text{arccsch}x=\ln\left(\frac{1}{x}+\frac{% \sqrt{1+x^{2}}}{\|x\|}\right)$

Let’s look at how to derive the first equation. The others follow similarly. Suppose $\displaystyle y=\sinh^{-1}x$ . Then, $\displaystyle x=\sinh y$ and, by the definition of the hyperbolic sine function, $\displaystyle x=\frac{e^{y}-e^{-y}}{2}$ . Therefore,

\displaystyle e^{y}-2x-e^{-y}=0.

Multiplying this equation by $\displaystyle e^{y}$ , we obtain

\displaystyle e^{2y}-2xe^{y}-1=0.

This can be solved like a quadratic equation, with the solution

\displaystyle e^{y}=\frac{2x\pm\sqrt{4x^{2}+4}}{2}=x\pm\sqrt{x^{2}+1}.

Since $\displaystyle e^{y}>0$ , the only solution is the one with the positive sign. Applying the natural logarithm to both sides of the equation, we conclude that

\displaystyle y=\ln\left(x+\sqrt{x^{2}+1}\right).

Example 3.10.2 (Evaluating Inverse Hyperbolic Functions).

Evaluate each of the following expressions.

\displaystyle\sinh^{-1}(2)

\displaystyle\tanh^{-1}(1/4)

Solution: $\displaystyle\sinh^{-1}(2)=\ln\left(2+\sqrt{2^{2}+1}\right)=\ln\left(2+\sqrt{5% }\right)\approx 1.4436$

$\displaystyle\tanh^{-1}(1/4)=\frac{1}{2}\ln\left(\frac{1+1/4}{1-1/4}\right)=% \frac{1}{2}\ln\left(\frac{5/4}{3/4}\right)=\frac{1}{2}\ln\left(\frac{5}{3}% \right)\approx 0.2554$

Checkpoint 3.10.2.

Evaluate $\displaystyle\tanh^{-1}(1/2)$ .

Hint: Use the definition of $\displaystyle\tanh^{-1}x$ and simplify.

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.

Table 3.3: Domains and Ranges of the Inverse Hyperbolic Functions

\displaystyle\begin{array}[]{|l|l|l|}\hline\cr\text{Function}&\text{Domain}&% \text{Range}\\ \hline\cr\sinh^{-1}x&(-\infty,\infty)&(-\infty,\infty)\\ \hline\cr\cosh^{-1}x&[1,\infty)&[0,\infty)\\ \hline\cr\tanh^{-1}x&(-1,1)&(-\infty,\infty)\\ \hline\cr\coth^{-1}x&(-\infty,-1)\cup(1,\infty)&(-\infty,0)\cup(0,\infty)\\ \hline\cr\text{sech}^{-1}x&(0,1]&[0,\infty)\\ \hline\cr\text{csch}^{-1}x&(-\infty,0)\cup(0,\infty)&(-\infty,0)\cup(0,\infty)% \\ \hline\cr\end{array}

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

Figure 3.31: Graphs of the inverse hyperbolic functions.

3.10.5 Derivatives of the Hyperbolic Functions

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

	$\displaystyle\frac{d}{dx}\left(\sinh x\right)$	$\displaystyle=\frac{d}{dx}\left(\frac{e^{x}-e^{-x}}{2}\right)$
		$\displaystyle=\frac{1}{2}\left[\frac{d}{dx}(e^{x})-\frac{d}{dx}(e^{-x})\right]$
		$\displaystyle=\frac{1}{2}\left[e^{x}+e^{-x}\right]=\cosh x.$

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

Table 3.4: Derivatives of the Hyperbolic Functions

\displaystyle\begin{array}[]{|l|l|}\hline\cr f(x)&\frac{d}{dx}f(x)\\[5.0pt] \hline\cr\sinh x&\cosh x\\ \hline\cr\cosh x&\sinh x\\ \hline\cr\tanh x&\mathop{\rm sech}\nolimits^{2}x\mathstrut\\ \hline\cr\coth x&-\mathop{\rm csch}\nolimits^{2}x\\ \hline\cr\mathop{\rm sech}x&-\mathop{\rm sech}x\tanh x\\ \hline\cr\mathop{\rm csch}x&-\mathop{\rm csch}x\coth x\\ \hline\cr\end{array}

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: $\displaystyle(d/dx)\sin x=\cos x$ and $\displaystyle(d/dx)\sinh x=\cosh x$ . The derivatives of the cosine functions, however, differ in sign: $\displaystyle(d/dx)\cos x=-\sin x$ , but $\displaystyle(d/dx)\cosh x=\sinh x$ . As we continue our examination of the hyperbolic functions, we must be mindful of their similarities and differences to the standard trigonometric functions.

Example 3.10.3 (Differentiating Hyperbolic Functions).

Evaluate the following derivatives:

(a)

$\displaystyle\frac{d}{dx}\left(\sinh(x^{2})\right)$
(b)

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

Solution: Using the formulas in Table 3.4 and the chain rule, we get

(a)

$\displaystyle\frac{d}{dx}\left(\sinh(x^{2})\right)=\cosh(x^{2})\cdot 2x$
(b)

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

Checkpoint 3.10.3.

Evaluate the following derivatives:

(a)

$\displaystyle\frac{d}{dx}\left(\tanh(x^{2}+3x)\right)$
(b)

$\displaystyle\frac{d}{dx}\left(\frac{1}{(\sinh x)^{2}}\right)$

Hint: Use the formulas in Table 3.4 and apply the chain rule as necessary.

3.10.6 Derivatives of Inverse Hyperbolic Functions

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

	$\displaystyle y$	$\displaystyle=\sinh^{-1}x$
	$\displaystyle\sinh y$	$\displaystyle=x$
	$\displaystyle\frac{d}{dx}\sinh y$	$\displaystyle=\frac{d}{dx}x$
	$\displaystyle\cosh y\frac{dy}{dx}$	$\displaystyle=1.$

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

\displaystyle\frac{dy}{dx}=\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.

Table 3.5: Derivatives of the Inverse Hyperbolic Functions

\displaystyle\begin{array}[]{|l|l|}\hline\cr f(x)&\frac{d}{dx}f(x)\\ \hline\cr\sinh^{-1}x&\frac{1}{\sqrt{1+x^{2}}}\\ \hline\cr\cosh^{-1}x&\frac{1}{\sqrt{x^{2}-1}}\\ \hline\cr\tanh^{-1}x&\frac{1}{1-x^{2}}\\ \hline\cr\coth^{-1}x&\frac{1}{1-x^{2}}\\ \hline\cr\text{sech}^{-1}x&\frac{-1}{x\sqrt{1-x^{2}}}\\ \hline\cr\text{csch}^{-1}x&\frac{-1}{|x|\sqrt{1+x^{2}}}\\ \hline\cr\end{array}

Example 3.10.4 (Differentiating Inverse Hyperbolic Functions).

Evaluate the following derivatives:

(a)

$\displaystyle\frac{d}{dx}\left(\sinh^{-1}\left(\frac{x}{3}\right)\right)$
(b)

$\displaystyle\frac{d}{dx}(\tanh^{-1}x)^{2}$

Solution: Using the formulas in Table 3.5 and the chain rule, we obtain the following results:

(a)

$\displaystyle\frac{d}{dx}\left(\sinh^{-1}\left(\frac{x}{3}\right)\right)=\frac% {1}{3\sqrt{1+\frac{x^{2}}{9}}}=\frac{1}{\sqrt{9+x^{2}}}$
(b)

$\displaystyle\frac{d}{dx}(\tanh^{-1}x)^{2}=\frac{2(\tanh^{-1}x)}{1-x^{2}}$

Checkpoint 3.10.4.

Evaluate the following derivatives:

(a)

$\displaystyle\frac{d}{dx}\left(\cosh^{-1}(3x)\right)$
(b)

$\displaystyle\frac{d}{dx}(\coth^{-1}x)^{3}$

Hint: Use the formulas in Table 3.5 and apply the chain rule as necessary.

Key Concepts

•

The hyperbolic functions involve combinations of the exponential functions $\displaystyle e^{x}$ and $\displaystyle e^{-x}$ . As a result, the inverse hyperbolic functions involve the natural logarithm.
•

Term-by-term differentiation yields differentiation formulas for the hyperbolic functions.
•

With appropriate range restrictions, the hyperbolic functions all have inverses.
•

Implicit differentiation yields differentiation formulas for the inverse hyperbolic functions.
•

The most common physical applications of hyperbolic functions are calculations involving catenaries.

Glossary

catenary: a curve in the shape of the function $\displaystyle y=a\cosh(x/a)$ is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary