6.9 Calculus of the hyperbolic functions

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Apply the formulas for derivatives and integrals of the hyperbolic functions.
Apply the formulas for the derivatives of the inverse hyperbolic functions and their associated integrals.
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. — 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.

Derivatives of the hyperbolic functions
$f (x)$	$\frac{d}{d x} f (x)$
$sinh x$	$cosh x$
$cosh x$	$sinh x$
$tanh x$	${sech}^{2} x$
$coth x$	$− {csch}^{2} x$
$sech x$	$− sech x tanh x$
$csch x$	$− csch x coth x$

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}

Differentiating hyperbolic functions

Evaluate the following derivatives:

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

Using the formulas in [link] and the chain rule, we get

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

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Evaluate the following derivatives:

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

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

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Integrals involving hyperbolic functions

Evaluate the following integrals:

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

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

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

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Evaluate the following integrals:

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

$\int {sinh}^{3} x cosh x d x = \frac{{sinh}^{4} x}{4} + C$
$\int {sech}^{2} (3 x) d x = \frac{tanh (3 x)}{3} + C$

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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.

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.

Kate Reply

To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.

Muhammed

What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?

Kate Reply

what is the change in momentum of a body?

Eunice Reply

what is a capacitor?

Raymond Reply

Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.

Gautam

A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate

Maria Reply

please solve

Sharon

8m/s²

Aishat

What is Thermodynamics

Muordit

velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.

Mehmet

A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat

Saheed Reply

50 m/s due south east

Someone

which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer

Ramon Reply

I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.

Someone

Scratch that

Someone

temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....

Someone

about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle

Someone

definitely of physics

Haryormhidey Reply

how many start and codon

Esrael Reply

what is field

Felix Reply

physics, biology and chemistry this is my Field

ALIYU

field is a region of space under the influence of some physical properties

Collete

what is ogarnic chemistry

WISDOM Reply

determine the slope giving that 3y+ 2x-14=0

WISDOM

Another formula for Acceleration

Belty Reply

a=v/t. a=f/m a

IHUMA

innocent

Adah

pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?

Nassze Reply

how do lnternal energy measures

Esrael

Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.

JALLAH Reply

No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.

Dlovan

Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?

Robert

like charges repel while unlike charges atttact

Raymond

What is specific heat capacity

Destiny Reply

Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).

AI-Robot

specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin

ROKEEB

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Source: OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2

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