Lesson 3 derivative of hyperbolic functions
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1. The document discusses differentiation formulas for hyperbolic functions including sinh, cosh, tanh, coth, sech, and csch. It provides examples of finding the derivatives of functions involving hyperbolic functions. 2. Hyperbolic functions are compared to trigonometric functions, noting that each pair of functions (e.g. sinh and cosh) are inverses of each other. Important hyperbolic identities are also listed. 3. Examples are given of finding the derivatives of functions involving hyperbolic functions, such as f(x) = xsinh(x). The document provides a concise review of differentiation rules for hyperbolic functions and examples of their application.
![A. Find the derivative of each of the following functions
and simplify the result:
x2coshxsinhy.1 =
)xcoshxsinh(xcosh'y
)xsinhx(coshxcosh)xcoshxsinh(xsinh'y
xcoshxcoshxsinhxsinh'y
22
22
5
22
222
+=
++=
+=
xhsecxy.2 =
xhsec)xtanhxhsec(x'y +−=
xhsecy.3 2
=
xtanhxhsecxhsec2'y −=
xhsecxcothy.5 =
)xhcsc(xhsec)xtanhxhsec(xcoth'y 2
−+−=
2
xsinhlny.4 =
2
2
xsinh
xcoshx2
'y =
EXAMPLE:
)xtanhx1(xhsec'y −=
xtanhxhsec2'y 2
−=
2
xcothx2'y =
[ ]xhcscxtanhxcothxhsec'y 2
+−=
[ ]xhcsc1xhsec'y 2
+−=
hxcscxcothy
xcothxhsec'y
'
−=
−= 2](https://image.slidesharecdn.com/lesson3derivativeofhyperbolicfunctions-150718183502-lva1-app6892/85/Lesson-3-derivative-of-hyperbolic-functions-7-320.jpg)
Lesson 3 derivative of hyperbolic functions
- 2. 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.
- 6. DIFFERENTIATION FORMULA Derivative of Hyperbolic Function
- 7. A. Find the derivative of each of the following functions and simplify the result: x2coshxsinhy.1 = )xcoshxsinh(xcosh'y )xsinhx(coshxcosh)xcoshxsinh(xsinh'y xcoshxcoshxsinhxsinh'y 22 22 5 22 222 += ++= += xhsecxy.2 = xhsec)xtanhxhsec(x'y +−= xhsecy.3 2 = xtanhxhsecxhsec2'y −= xhsecxcothy.5 = )xhcsc(xhsec)xtanhxhsec(xcoth'y 2 −+−= 2 xsinhlny.4 = 2 2 xsinh xcoshx2 'y = EXAMPLE: )xtanhx1(xhsec'y −= xtanhxhsec2'y 2 −= 2 xcothx2'y = [ ]xhcscxtanhxcothxhsec'y 2 +−= [ ]xhcsc1xhsec'y 2 +−= hxcscxcothy xcothxhsec'y ' −= −= 2
- 8. xcothlny.6 2 = xcoth xhcscxcoth2 'y 2 2 − = xcotharccosy.7 = xcoth1 xhcsc 'y 2 2 − − −= xhcscxhcsc xhcscxhcsc 'y 22 22 −•− −• = xsinh xcosh xsinh 1 2 'y 2 − = 2 2 xsinhxcosh 2 'y • − = x2sinh 4 'y −= x2hcsc4'y −= xhcsc xhcsc 'y 2 2 − = xhcsc'y 2 −=
- 9. )xharctan(siny.8 2 = ( )22 2 xcosh xcoshx2 'y = 2 xhsecx2'y = ( )22 2 xsinh1 xcoshx2 'y + =
- 10. A. Find the derivative and simplify the result. ( ) 2 xsinhxf.1 = ( ) w4hsecwF.2 2 = ( ) 3 xtanhxG.3 = ( ) 3 tcoshtg.4 = ( ) x 1 cothxh.5 = ( ) ( )xtanhlnxg.6 = EXERCISES: ( ) ( )ylncothyf.7 = ( ) xcoshexh.8 x = ( ) ( )x2sinhtanxf.9 1− = ( ) ( )x xsinhxg.10 = ( ) ( )21 xtanhsinxg.11 − = ( ) 0x,xxf.12 xsinh >=
- 11. Hyperbolic Functions Trigonometric Functions 1xsinhxcosh 22 =− xhsecxtanh1 22 =− xhcsc1xcoth 22 =− ysinhxcoshycoshxsinh)yxsinh( ±=± ( ) ysinhxsinhycoshxcoshyxcosh ±=± ( ) ytanhxtanh1 ytanhxtanh yxtanh ± ± =± ( ) ytanxtan1 ytanxtan yxtan ± =± ( ) ysinxsinycosxcosyxcos =± ( ) ysinxcosycosxsinyxsin ±=± xx 22 sectan1 =+ 1sincos 22 =+ xx xcsc1xcot 22 =+ Identities: Hyperbolic Functions vs. Trigonometric Functions
- 12. Hyperbolic Functions Trigonometric Functions Identities: Hyperbolic Functions vs. Trigonometric Functions sinh 2x = 2 sinh x cosh x ( ) 2/1x2coshxsinh2 −= ( ) 2/1x2coshxcosh2 += x exsinhxcosh =+ x exsinhxcosh − =− ( ) 2/x2cos1xcos2 += ( ) 2/x2cos1xsin2 −= cos 2x = cos2 x – sin2 x sin 2x = 2sinx cosx cosh 2x = cosh2 x +sinh2 x