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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