# 1.5 Exponential and logarithmic functions  (Page 6/17)

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Compare the relative severity of a magnitude $8.4$ earthquake with a magnitude $7.4$ earthquake.

The magnitude $8.4$ earthquake is roughly $10$ times as severe as the magnitude $7.4$ earthquake.

## Hyperbolic functions

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

## Definition

Hyperbolic cosine

$\text{cosh}\phantom{\rule{0.1em}{0ex}}x=\frac{{e}^{x}+{e}^{\text{−}x}}{2}$

Hyperbolic sine

$\text{sinh}\phantom{\rule{0.1em}{0ex}}x=\frac{{e}^{x}-{e}^{\text{−}x}}{2}$

Hyperbolic tangent

$\text{tanh}\phantom{\rule{0.1em}{0ex}}x=\frac{\text{sinh}\phantom{\rule{0.1em}{0ex}}x}{\text{cosh}\phantom{\rule{0.1em}{0ex}}x}=\frac{{e}^{x}-{e}^{\text{−}x}}{{e}^{x}+{e}^{\text{−}x}}$

Hyperbolic cosecant

$\text{csch}\phantom{\rule{0.1em}{0ex}}x=\frac{1}{\text{sinh}\phantom{\rule{0.1em}{0ex}}x}=\frac{2}{{e}^{x}-{e}^{\text{−}x}}$

Hyperbolic secant

$\text{sech}\phantom{\rule{0.1em}{0ex}}x=\frac{1}{\text{cosh}\phantom{\rule{0.1em}{0ex}}x}=\frac{2}{{e}^{x}+{e}^{\text{−}x}}$

Hyperbolic cotangent

$\text{coth}\phantom{\rule{0.1em}{0ex}}x=\frac{\text{cosh}\phantom{\rule{0.1em}{0ex}}x}{\text{sinh}\phantom{\rule{0.1em}{0ex}}x}=\frac{{e}^{x}+{e}^{\text{−}x}}{{e}^{x}-{e}^{\text{−}x}}$

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 $\text{cosh}\left(x\right)$ and principles of physics, it can be shown that the height of a hanging chain, such as the one in [link] , can be described by the function $h\left(x\right)=a\phantom{\rule{0.1em}{0ex}}\text{cosh}\left(x\text{/}a\right)+c$ for certain constants $a$ and $c.$

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

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

This identity is the analog of the trigonometric identity ${\text{cos}}^{2}t+{\text{sin}}^{2}t=1.$ Here, given a value $t,$ the point $\left(x,y\right)=\left(\text{cosh}\phantom{\rule{0.1em}{0ex}}t,\text{sinh}\phantom{\rule{0.1em}{0ex}}t\right)$ lies on the unit hyperbola ${x}^{2}-{y}^{2}=1$ ( [link] ).

## Graphs of hyperbolic functions

To graph $\text{cosh}\phantom{\rule{0.1em}{0ex}}x$ and $\text{sinh}\phantom{\rule{0.1em}{0ex}}x,$ we make use of the fact that both functions approach $\left(1\text{/}2\right){e}^{x}$ as $x\to \infty ,$ since ${e}^{\text{−}x}\to 0$ as $x\to \infty .$ As $x\to \text{−}\infty ,\text{cosh}\phantom{\rule{0.1em}{0ex}}x$ approaches $1\text{/}2{e}^{\text{−}x},$ whereas $\text{sinh}\phantom{\rule{0.1em}{0ex}}x$ approaches $-1\text{/}2{e}^{\text{−}x}.$ Therefore, using the graphs of $1\text{/}2{e}^{x},1\text{/}2{e}^{\text{−}x},$ and $\text{−}1\text{/}2{e}^{\text{−}x}$ as guides, we graph $\text{cosh}\phantom{\rule{0.1em}{0ex}}x$ and $\text{sinh}\phantom{\rule{0.1em}{0ex}}x.$ To graph $\text{tanh}\phantom{\rule{0.1em}{0ex}}x,$ we use the fact that $\text{tanh}\left(0\right)=1,-1<\text{tanh}\left(x\right)<1$ for all $x,\text{tanh}\phantom{\rule{0.1em}{0ex}}x\to 1$ as $x\to \infty ,$ and $\text{tanh}\phantom{\rule{0.1em}{0ex}}x\to \text{−}1$ as $x\to \text{−}\infty .$ The graphs of the other three hyperbolic functions can be sketched using the graphs of $\text{cosh}\phantom{\rule{0.1em}{0ex}}x,\text{sinh}\phantom{\rule{0.1em}{0ex}}x,$ and $\text{tanh}\phantom{\rule{0.1em}{0ex}}x$ ( [link] ).

## Identities involving hyperbolic functions

The identity ${\text{cosh}}^{2}t-{\text{sinh}}^{2}t,$ shown in [link] , 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: identities involving hyperbolic functions

1. $\text{cosh}\left(\text{−}x\right)=\text{cosh}\phantom{\rule{0.1em}{0ex}}x$
2. $\text{sinh}\left(\text{−}x\right)=\text{−}\text{sinh}\phantom{\rule{0.1em}{0ex}}x$
3. $\text{cosh}\phantom{\rule{0.1em}{0ex}}x+\text{sinh}\phantom{\rule{0.1em}{0ex}}x={e}^{x}$
4. $\text{cosh}\phantom{\rule{0.1em}{0ex}}x-\text{sinh}\phantom{\rule{0.1em}{0ex}}x={e}^{\text{−}x}$
5. ${\text{cosh}}^{2}x-{\text{sinh}}^{2}x=1$
6. $1-{\text{tanh}}^{2}x={\text{sech}}^{2}x$
7. ${\text{coth}}^{2}x-1={\text{csch}}^{2}x$
8. $\text{sinh}\left(x±y\right)=\text{sinh}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{cosh}\phantom{\rule{0.1em}{0ex}}y±\text{cosh}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{sinh}\phantom{\rule{0.1em}{0ex}}y$
9. $\text{cosh}\left(x±y\right)=\text{cosh}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{cosh}\phantom{\rule{0.1em}{0ex}}y±\text{sinh}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{sinh}\phantom{\rule{0.1em}{0ex}}y$

The f'(4)for f(x) =4^x
I need help under implicit differentiation
how to understand this
nhie
What is derivative of antilog x dx ?
what's the meaning of removable discontinuity
what's continuous
Brian
an area under a curve is continuous because you are looking at an area that covers a range of numbers, it is over an interval, such as 0 to 4
Lauren
using product rule x^3,x^5
Kabiru
may god be with you
Sunny
Luke 17:21 nor will they say, See here or See there For indeed, the kingdom of God is within you. You've never 'touched' anything. The e-energy field created by your body has pushed other electricfields. even our religions tell us we're the gods. We live in energies connecting us all. Doa/higgsfield
Scott
if you have any calculus questions many of us would be happy to help and you can always learn or even invent your own theories and proofs. math is the laws of logic and reality. its rules are permanent and absolute. you can absolutely learn calculus and through it better understand our existence.
Scott
ya doubtless
Bilal
help the integral of x^2/lnxdx
Levis
also find the value of "X" from the equation that follow (x-1/x)^4 +4(x^2-1/x^2) -6=0 please guy help
Levis
Use integration by parts. Let u=lnx and dv=x2dx Then du=1xdx and v=13x3. ∫x2lnxdx=13x3lnx−∫(13x3⋅1x)dx ∫x2lnxdx=13x3lnx−∫13x2dx ∫x2lnxdx=13x3lnx−19x3+C
Bilal
itz 1/3 and 1/9
Bilal
now you can find the value of X from the above equation easily
Bilal
Pls i need more explanation on this calculus
usman
usman from where do you need help?
Levis
thanks Bilal
Levis
integrate e^cosx
Uchenna
-sinx e^x
Leo
Do we ask only math question? or ANY of the question?
yh
Gbesemete
How do i differentiate between substitution method, partial fraction and algebraic function in integration?
usman
you just have to recognize the problem. there can be multiple ways to solve 1 problem. that's the hardest part about integration
Lauren
test
we asking the question cause only the question will tell us the right answer
Sunny
find integral of sin8xcos12xdx
don't share these childish questions
Bilal
well find the integral of x^x
Levis
bilal kumhar you are so biased if you are an expert what are you doing here lol😎😎😂😂 we are here to learn and beside there are many questions on this chat which you didn't attempt we are helping each other stop being naive and arrogance so give me the integral of x^x
Levis
Levis I am sorry
Bilal
Bilal it okay buddy honestly i am pleasured to meet you
Levis
x^x ... no anti derivative for this function... but we can find definte integral numerically.
Bilal
thank you Bilal Kumhar then how we may find definite integral let say x^x,3,5?
Levis
evaluate 5-×square divided by x+2 find x as limit approaches infinity
i have not understood
Leo
Michael
welcome
Sunny
I just dont get it at all...not understanding
Michagaye
0 baby
Sunny
The denominator is the aggressive one
Sunny
wouldn't be any prime number for x instead ?
Harold
or should I say any prime number greater then 11 ?
Harold
just wondering
Harold
I think as limit Approach infinity then X=0
Levis
ha hakdog hahhahahaha
ha hamburger
Leonito
Fond the value of the six trigonometric function of an angle theta, which terminal side passes through the points(2x½-y)²,4
What's f(x) ^x^x
What's F(x) =x^x^x
Emeka
are you asking for the derivative
Leo
that's means more power for all points
rd
Levis
iam sorry f(x)=x^x it means the output(range ) depends to input(domain) value of x by the power of x that is to say if x=2 then x^x would be 2^2=4 f(x) is the product of X to the power of X its derivatives is found by using product rule y=x^x introduce ln each side we have lny=lnx^x =lny=xlnx
Levis
the derivatives of f(x)=x^x IS (1+lnx)*x^x
Levis
So in that case what will be the answer?
Alice
nice explanation Levis, appreciated..
Thato
what is a maximax
A maxima in a curve refers to the maximum point said curve. The maxima is a point where the gradient of the curve is equal to 0 (dy/dx = 0) and its second derivative value is a negative (d²y/dx² = -ve).
Viewer
what is the limit of x^2+x when x approaches 0
it is 0 because 0 squared Is 0
Leo
0+0=0
Leo
simply put the value of 0 in places of x.....
Tonu
the limit is 2x + 1
Nicholas
the limit is 0
Muzamil
limit s x
Bilal
The limit is 3
Levis
Leo we don't just do like that buddy!!! use first principle y+∆y=x+∆x ∆y=x+∆x-y ∆y=(x+∆x)^2+(x+∆x)-x^2+x on solving it become ∆y=3∆x+∆x^2 as ∆x_>0 limit=3 if you do by calculator say plugging any value of x=0.000005 which approach 0 you get 3
Levis
find derivatives 3√x²+√3x²
3 + 3=6
mujahid
How to do basic integrals
the formula is simple x^n+1/n+1 where n IS NOT EQUAL TO 1 And n stands for power eg integral of x^2 x^2+1/2+1 =X^3/3
Levis