Upload
lawrence-de-vera
View
93
Download
5
Embed Size (px)
Citation preview
TRANSCENDENTAL FUNCTIONS
Kinds of transcendental functions:1.logarithmic and exponential functions2.trigonometric and inverse trigonometric functions3.hyperbolic and inverse hyperbolic functions Note:Each pair of functions above is an inverse to each other.
A. Find the derivative of each of the following functions and simplify the result:
x2 cosh x sinhy .1
)xcoshxsinh(xcosh'y
)xsinhx(coshxcosh)xcoshxsinh(xsinh'y
x cosh x coshx sinh x sinh'y
22
22
5
22
222
x hsecxy .2 x hsec)x tanhx hsec(x'y
xhsecy .3 2
x tanhx hsecx hsec2'y
x hsecx cothy .5
)xhcsc(x hsec)x tanhx hsec(xcoth'y 2
2x sinhlny .4
2
2
xsinh
xcosh x2'y
EXAMPLE:
)x tanh x1(x hsec'y
x tanhxhsec2'y 2
2xcoth x2'y
xhcscx tanhx cothx hsec'y 2
xhcsc1x hsec'y 2
hxcscxcothy
xcothx hsec'y'
2
xcoth lny .6 2
xcoth
x hcsc x coth2'y
2
2
xcotharccosy .7
xcoth1
xhcsc'y
2
2
xhcscxhcsc
xhcscxhcsc'y
22
22
x sinh
x coshxsinh
12
'y2
2
2
x sinhxcosh
2'y
x2 sinh
4'y
x2 hcsc4'y
xhcsc
xhcsc'y
2
2
xhcsc'y 2
A. Find the derivative and simplify the result.
2xsinhxf .1
w4hsecwF .2 2
3 xtanhxG .3
3tcoshtg .4
x
1cothxh .5
xtanhlnxg .6
EXERCISES:
ylncothyf .7
xcoshexh .8 x
x2sinhtanxf .9 1
xxsinhxg .10
21 xtanhsinxg .11
0x,xxf .12 xsinh
Hyperbolic Functions Trigonometric Functions
1xsinhxcosh 22
xhsecxtanh1 22
xhcsc1xcoth 22
ysinhxcoshycoshxsinh)yxsinh(
ysinhxsinhycoshxcoshyxcosh
ytanhxtanh1
ytanhxtanhyxtanh
ytanxtan1
ytanxtanyxtan
ysinxsinycosxcosyxcos
ysinxcosycosxsinyxsin
xx 22 sectan1
1sincos 22 xx
xcsc1xcot 22
Identities: Hyperbolic Functions vs. Trigonometric Functions
Hyperbolic Functions Trigonometric Functions
Identities: Hyperbolic Functions vs. Trigonometric Functions
sinh 2x = 2 sinh x cosh x
2/1x2coshxsinh2
2/1x2coshxcosh2 xexsinhxcosh
xexsinhxcosh
2/x2cos1xcos2
2/x2cos1xsin 2
cos 2x = cos2x – sin2x
sin 2x = 2sinx cosx
cosh 2x = cosh2x +sinh2x