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Week 7
Inverse trigonometric functions
Hyperbolic functions
Inverse hyperbolic functions
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2Copyright Cengage Learning. ll rights reserve!.
6.6 Inverse "rigonometric #unctions
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$
Inverse "rigonometric #unctions
"he !ifferentiation formulas for all of the inversetrigonometric functions.
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&'ample 1
If f(x) * sin+1(x2+ 1), fin! f(x).
( ') *
f-(') * . ()
* . 2' *
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&'ample 2
/ifferentiate (a) y*
0olution
y * ( ') *
y-
Chain rule
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b) f(x) *x arctan
3ro!uct rule
' *
&'ample 2 (b). #in! f-(')
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&'ercise 1
Calculate the !erivative of ()
' *
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&'ercise 2
Calculate the !erivative of ' $'
( ') *
3ro!uct rule
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5Copyright Cengage Learning. ll rights reserve!.
6.7 Hyperbolic #unctions
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Hyperbolic #unctions
#or this reason they are collectively calle! hyperbolicfunctions an! in!ivi!ually calle! hyperbolic sine,
hyperbolic cosine, an! so on.
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Hyperbolic #unctions
"he hyperbolic functions satisfy a number of i!entities thatare similar to ell8knon trigonometric i!entities.
We list some of them here.
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Hyperbolic #unctions
"he !erivatives of the hyperbolic functions are easilycompute!. #or e'ample,
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Hyperbolic #unctions
We list the !ifferentiation formulas for the hyperbolicfunctions as "able 1.
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&'ample $
ny of these !ifferentiation rules can be combine! ith theChain 9ule. #or instance,
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&'ercise $
y * . #in! y- :
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&'ercise %
y * sin h (cosh '). #in! y- :
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Inverse Hyperbolic #unctions
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Inverse Hyperbolic #unctions
"he inverse hyperbolic functions are all !ifferentiablebecause the hyperbolic functions are !ifferentiable.
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' *
0ho that ; (sin ')< * sec '
&'ample %
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&'ercise
( ') *
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