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DIFFERENTIATION OF

INVERSE HYPERBOLICFUNCTIONS

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OBJECTIVES: define and graph inverse hyperbolic

functions;

prove some exercises on the logarithmicequivalents of inverse hyperbolic functions

differentiate inverse hyperbolic functions.

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TRANSCENDENTAL FUNCTIONS

Kinds of transcendental functions:

1. logarithmic and exponential functions

2. trigonometric and inverse trigonometric

functions

3. hyperbolic and inverse hyperbolic functions

Note:

Each pair of functions above is an inverse to

each other.

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Table 6.9.1 (p. 479)

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Figure 6.9.6 (p. 479) Graphs of inverse hyperbolic functions

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Theorem 6.9.4 (p. 480)

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Theorem 6.9.5 (p. 481)

DIFFERENTIATION FORMULA

Derivative of Inverse Hyperbolic Function

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A. Find the derivative of each of the following functions

and simplify the result:

EXAMPLE:

21 x1xsinhxxG.1

2

1

2x12

x21xsinh1

x1

1xx'G

2

1

2x1

xxsinh1x

xx'G

xsinhx'G 1212 xcoshxy.2

x2xcoshx21x

1x'y 21

4

2

214

2

xcosh1x

xx2

21424

4

xcosh1xx1x

1xx2'y

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xtanhx1xlny.3 12

1xtanh1x1

1x

1x2

x2

1x

1'y

1

222

xtanhx1

x

1x

x'y

1

22

xtanh'y 1

313 x31cothlogxF.4

313 x31cothlog2

1xF

223

331 x9x311

1elog

x31coth

1

2

1x'F

3163

3

2

x31cothx9x6112

elogx9x'F

3133

3

2

x31cothx32x32

elogx9x'F

313

3

x31cothx32x2

elog3x'F

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x4cothcothxG.5 1

x424

x4hcscx4coth1

1x'G 22

x4hcscx4x4hcsc2

x'G2

2

xx

x22x'G

2

xhcscy.6

21

x221

2

x12

x

1'y222

44

2 x4x

4

4x4x

x2'y

44

x4x

x44'y

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A. Find the derivative and simplify the result.

EXERCISES:

21 x3tansinhxf.1

x

1csccosh3xh.2 1

2x31

ecostanhy.3

2i x6hsechsecxg.4

21xhsecy.5

2x95 esinhlogy.6

22 x31coshx31sinh3xh.7

xcoshxsinhxG.8

1x3cosh11x3sinh

xH.9

3

x

1coshxF.10

2x5tanhlnxf.11