Lesson 1 Derivative of Trigonometric Functions
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Transcript of Lesson 1 Derivative of Trigonometric Functions
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DIFFERENTIATION OF TRIGONOMETRIC
FUNCTIONS
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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.
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The TRIGONOMETRIC FUNCTIONS.
xtan
1
xsin
xcosxcot .4
xcot
1
xcos
xsinxtan .3
x cos
1 x sec
xsec
1xcos 2.
x sin
1x csc
xcsc
1xsin 1.
Identities ciprocalRe .A
Identities ricTrigonomet
:callRe
ytanxtan1
ytanxtanyxtan .3
ysinxsinycosxcosyxcos 2.
ysinxcosycosxsinyxsin 1.
Angles Two of Difference and Sum.B
xtan1
xtan2x2tan .3
1xcos2
xsin21
xsinxcosx2cos 2.
2sinxcosx x2sin 1.
Formulas Angle Double .C
2
2
2
22
xcscxcot1 .3
xsecxtan1 .2
1xcosxsin .1
Identities Squared .D
22
22
22
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DIFFERENTIATION FORMULADerivative of Trigonometric FunctionFor the differentiation formulas of the trigonometric functions, all you need to know is the differentiation formulas of sin u and cos u. Using these formulas and the differentiation formulas of the algebraic functions, the differentiation formulas of the remaining functions, that is, tan u, cot u, sec u and csc u may be obtained.
dx
duusinucos
dx
d
dx
duucosusin
dx
d
xfu where u cos of Derivative
xfu where u sin of Derivative
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xcos
xsin
dx
d xtan
dx
d
xfu where u tan of Derivative
2cosx
xcosdxd
sinxxsindxd
cosx xtan
dx
d
quotient, of derivative gsinU
xcos
xsinsinxcosxcosx
2
xcos
1
xcos
xsinxcos
22
22
x secxtandx
d 2
dxdu
usec utandxd
Therefore 2
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xtan
1
dx
d xcot
dx
d
xfu where u cot of Derivative
2
2
2 tanx
xsec10
tanx
xtandxd
10 xtan
dx
d
quotient, of derivative gsinU
xcsc xsin
1
xcosxsinxcos
1
xtan
xsec 2
2
2
2
2
2
2
xcsc- xcotdx
d 2
dxdu
ucsc- ucotdxd
Therefore 2
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xcos
1
dx
d xsec
dx
d
xfu where u sec of Derivative
22 cosx
xsin10
cosx
xcosdxd
10 xtan
dx
d
quotient, of derivative gsinU
x secx tan xcos
1
xcos
xsin
xcos
xsin
2
x secx tan xsecdx
d
dxdu
usecutan usecdxd
Therefore
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xsin
1
dx
d xcsc
dx
d
xfu where u csc of Derivative
22 x sin
xcos10
x sin
xsindxd
10 xcsc
dx
d
quotient, of derivative gsinU
x csc x cot xsin
1
xsin
xcos
xsin
xcos
2
x csc x cot xcscdx
d
dxdu
ucscucot- ucscdxd
Therefore
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dx
du u cosu sin
dx
d
dx
du u sinu cos
dx
d
dx
du usecu tan
dx
d 2
dx
du ucscu cot
dx
d 2
dx
du u sec u tanu sec
dx
d
dx
du u csc u cotu csc
dx
d
If u is a differentiable function of x, then the following are differentiation formulas of the trigonometric functions
SUMMARY:
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A. Find the derivative of each of the following functions and simplify the result:
x3sin2xf .1
xsinexg .2
22 x31cosxh .3
x3cos6
3x3cos2x'f
xsindx
dex'g xsin
22x31cosxh
x2
1xcose xsin
x2
xcose x
x
x
x2
xcosex'g
xsinxsin
x6x31sinx31cos2x'h 22
22 x31sinx31cos2x6
2sinxcosx2xsinfrom
2x312sinx6x'h
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3x4cos3x4sin3y .4
233233 x12x4cosx4cosx12x4sinx4sin3'y
xsinxcos2xcos from 22
32 x42cosx36'y
32 x8cosx36'y
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x2
xtan2xf .5
12
1
2
xsec2x'f 2
12
xsecx'f 2
2
xtanx'f 2
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x1
xtan3logxh .6
22
3x1
1x1x1
x1
xsecelog
x1x
tan
1x'h
x1
xcos
1
x1
xsin
x1
xcos
x1
xx1elogx'h
22
3
2
2
x1
xcos
x1
xsin
1
x1
elogx'h
23
x1
xcos
x1
xsin2
1
x1
elog2x'h
23
x1
x2sin
1
x1
elog2x'h
23
x1
x2csc
x1
elog2x'h
23
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x2cosx2secy .7
x2seclnx2cosyln
x2seclnyln
sidesboth on ln Applyx2cos
2x2sinx2secln2x2tanx2secx2sec
1x2cos'y
y
1
ationdifferenti clogarithmi By
x2seclnx2sin2x2cos
x2sin2x2cos'y
y
1
x2secln1x2sin2
yx2secln1x2sin2'y
x2cosx2secx2secln1x2sin2'y
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xcot1
xcot2xh .8
2
22
222
xcot1
1xcscxcot2xcot21xcsc2xcot1x'h
xcot1xcot2xcot1
xcsc2x'h 22
22
2
1xcotxcsc
xcsc2 222
2
1
xsin
xcosxsin2
xcsc
1xcot2x'h
2
22
2
2
xsin
xsinxcosxsin2x'h
2
222
x2cos2x'h
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1xcscxF .9 3
1xcsc2
x31xcot1xcscx'F
3
233
1xcsc2
1xcsc1xcot1xcscx3x'F
3
3332
1xcsc1xcotx2
3x'F 332
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Find the derivative and simplify the result.
3x45sinlnxh .1
3 2xlncosxf .2
x4cos2
x4sinxg .3
x2cosx4sin2x2sinxcos2xF .4
xcos31
siny .5
3
x tanxsinxF . 6
yxtany .7
2
2
x1
x2cotxF .8
0xyxycot .9
EXERCISES:
0ycscxsec .10 22