Lesson 1 derivative of trigonometric functions
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DIFFERENTIATION OF TRIGONOMETRIC
FUNCTIONS
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.
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
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
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
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
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
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
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:
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
3x4cos3x4sin3y .4
233233 x12x4cosx4cosx12x4sinx4sin3'y
xsinxcos2xcos from 22
32 x42cosx36'y
32 x8cosx36'y
x2
xtan2xf .5
12
1
2
xsec2x'f 2
12
xsecx'f 2
2
xtanx'f 2
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
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
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
1xcscxF .9 3
1xcsc2
x31xcot1xcscx'F
3
233
1xcsc2
1xcsc1xcot1xcscx3x'F
3
3332
1xcsc1xcotx2
3x'F 332
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
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