lesson 11 derivative of trigonometric functions

DIFFERENTIATION OF TRIGONOMETRIC

FUNCTIONS

OBJECTIVES:• differentiate functions involving trigonometric

functions;• solve problems involving differentiation of

trigonometric functions; and• apply trigonometric identities to simplify the

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.

xtan1

xsinxcosxcot .4

xcot1

xcosxsinxtan .3

x cos1 x sec

xsec1xcos 2.

x sin1x csc

xcsc1xsin 1.

Identities ciprocalRe .AIdentities ricTrigonomet

:callRe

ytanxtan1

ytanxtanyxtan .3

ysinxsinycosxcosyxcos 2. ysinxcosycosxsinyxsin 1.

Angles Two of Difference and Sum.B

xtan1xtan2x2tan .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.

dxduusinucos

dxd

dxduucosusin

dxd

xfu where u cos of Derivative

xfu where u sin of Derivative

xcosxsin

dxd xtan

dxd

xfu where u tan of Derivative

2cosx

xcosdxdsinxxsin

dxdcosx

xtandxd

quotient, of derivative gsinU

xcos

xsinsinxcosxcosx 2

xcos1

xcosxsinxcos 22

22

x secxtandxd 2

dxduusec utandx

d Therefore 2

xtan1

dxd xcot

dxd

xfu where u cot of Derivative

2

2

2 tanxxsec10

tanx

xtandxd10

xtandxd

quotient, of derivative gsinU

xcsc xsin

1

xcosxsinxcos

1

xtanxsec 2

2

2

2

2

2

2

xcsc- xcotdxd 2

dxduucsc- ucotdx

d Therefore 2

xcos1

dxd xsec

dxd

xfu where u sec of Derivative

22 cosx

xsin10 cosx

xcosdxd10

xtandxd

quotient, of derivative gsinU

x secx tan xcos

1xcosxsin

xcosxsin 2

x secx tan xsecdxd

dxduusecutan usecdx

d Therefore

xsin1

dxd xcsc

dxd

xfu where u csc of Derivative

22 x sin

xcos10 x sin

xsindxd10

xcscdxd

quotient, of derivative gsinU

x csc x cot xsin

1xsinxcos

xsinxcos 2

x csc x cot xcscdxd

dxduucscucot- ucscdx

d Therefore

dxdu u cosu sin

dxd

dxdu u sinu cos

dxd

dxdu usecu tan

dxd 2

dxdu ucscu cot

dxd 2

dxdu u sec u tanu sec

dxd

dxdu u csc u cotu csc

dxd

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

xsindxdex'g xsin

22x31cosxh

x21xcose xsin

x2

xcose xxx

x2xcosex'g

xsinxsin

x6x31sinx31cos2x'h 22

22 x31sinx31cos2x6

2sinxcosx2xsinfrom

2x312sinx6x'h

3x4cos3x4sin3y .4

233233 x12x4cosx4cosx12x4sinx4sin3'y

xsinxcos2xcos from 22

32 x42cosx36'y

32 x8cosx36'y

x2xtan2xf .5

121

2xsec2x'f 2

12xsecx'f 2

2xtanx'f 2

x1

xtan3logxh .6

22

3 x11x1x1

x1xsecelog

x1xtan

1x'h

x1xcos

1

x1xsin

x1xcos

x1xx1elogx'h

22

3

2

2

x1xcos

x1xsin

1x1

elogx'h 2

3

x1xcos

x1xsin2

1x1

elog2x'h 2

3

x1x2sin

1x1

elog2x'h 2

3

x1x2csc

x1elog2

x'h 23

x2cosx2secy .7

x2seclnx2cosyln

x2seclnyln

sidesboth on ln Applyx2cos

2x2sinx2secln2x2tanx2secx2sec

1x2cos'yy1

ationdifferenti clogarithmi By

x2seclnx2sin2x2cosx2sin2x2cos'y

y1

x2secln1x2sin2

yx2secln1x2sin2'y

x2cosx2secx2secln1x2sin2'y

xcot1

xcot2xh .8 2

22

222

xcot11xcscxcot2xcot21xcsc2xcot1x'h

xcot1xcot2xcot1

xcsc2x'h 2222

2

1xcotxcsc

xcsc2 222

2

1

xsinxcosxsin2

xcsc1xcot2x'h 2

22

2

2

xsinxsinxcosxsin2x'h 2

222

x2cos2x'h

1xcscxF .9 3

1xcsc2

x31xcot1xcscx'F3

233

1xcsc2

1xcsc1xcot1xcscx3x'F 3

3332

1xcsc1xcotx23x'F 332

Find the derivative and simplify the result.

3x45sinlnxh .1

3 2xlncosxf .2

x4cos2

x4sinxg .3

x2cosx4sin2x2sinxcos2xF .4

xcos31siny .5

3

x tanxsinxF .6

yxtany .7

2

2

x1x2cotxF .8

0xyxycot .9

EXERCISES:

0ycscxsec .10 22