Lesson 1 derivative of trigonometric functions

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