Lesson 1 derivative of trigonometric functions
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Lesson 1 derivative of trigonometric functions
- 2. TRANSCENDENTAL FUNCTIONS Kinds of transcendental functions: 1.logarithmic and exponential functions 2.trigonometric and inverse trigonometric functions 3.hyperbolic and inverse hyperbolic functions Note: Each pair of functions above is an inverse to each other.
- 3. The TRIGONOMETRIC FUNCTIONS . xtan 1 xsin xcos xcot.4 xcot 1 xcos xsin xtan.3 xcos 1 xsec xsec 1 xcos2. xsin 1 xcsc xcsc 1 xsin1. IdentitiesciprocalRe.A IdentitiesricTrigonomet :callRe == == =⇔= =⇔= ( ) ( ) ( ) ytanxtan1 ytanxtan yxtan.3 ysinxsinycosxcosyxcos2. ysinxcosycosxsinyxsin1. AnglesTwoofDifferenceandSum.B ± =± =± ±=± xtan1 xtan2 x2tan.3 1xcos2 xsin21 xsinxcosx2cos2. 2sinxcosxx2sin1. FormulasAngleDouble.C 2 2 2 22 − = −= −= −= = xcscxcot1.3 xsecxtan1.2 1xcosxsin.1 IdentitiesSquared.D 22 22 22 =+ =+ =+
- 4. DIFFERENTIATION FORMULA Derivative of Trigonometric Function For 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 du usinucos dx d dx du ucosusin dx d −= = = = xfuwhereucosofDerivative xfuwhereusinofDerivative
- 5. ( ) ( ) = = xcos xsin dx d xtan dx d xfuwhereutanofDerivative ( ) ( ) ( ) ( ) ( ) ( )2 cosx xcos dx d sinxxsin dx d cosx xtan dx d quotient,ofderivativegsinU − = ( )( ) ( )( ) xcos xsinsinxcosxcosx 2 −− = xcos 1 xcos xsinxcos 22 22 = + = ( ) xsecxtan dx d 2 = ( ) dx du usecutan dx d Therefore 2 =
- 6. ( ) ( ) = = xtan 1 dx d xcot dx d xfuwhereucotofDerivative ( ) ( ) ( ) ( ) ( ) ( )2 2 2 tanx xsec10 tanx xtan dx d 10 xtan dx d quotient,ofderivativegsinU − = − = xcsc xsin 1 xcos xsin xcos 1 xtan xsec 2 2 2 2 2 2 2 −= − = − = − = ( ) xcsc-xcot dx d 2 = ( ) dx du ucsc-ucot dx d Therefore 2 =
- 7. ( ) ( ) = = xcos 1 dx d xsec dx d xfuwhereusecofDerivative ( ) ( ) ( ) ( ) ( )( ) ( )22 cosx xsin10 cosx xcos dx d 10 xtan dx d quotient,ofderivativegsinU −− = − = xsecxtan xcos 1 xcos xsin xcos xsin 2 =⋅= + = ( ) xsecxtanxsec dx d = ( ) dx du usecutanusec dx d Therefore =
- 8. ( ) ( ) = = xsin 1 dx d xcsc dx d xfuwhereucscofDerivative ( ) ( ) ( ) ( ) ( )( ) ( )22 xsin xcos10 xsin xsin dx d 10 xcsc dx d quotient,ofderivativegsinU − = − = xcscxcot xsin 1 xsin xcos xsin xcos 2 −=⋅ − = − = ( ) xcscxcotxcsc dx d −= ( ) dx du ucscucot-ucsc dx d Therefore =
- 9. ( ) dx du ucosusin dx d = ( ) dx du usinucos dx d −= ( ) dx du usecutan dx d 2 = ( ) dx du ucscucot dx d 2 −= ( ) dx du usecutanusec dx d = ( ) dx du ucscucotucsc dx d −= If u is a differentiable function of x, then the following are differentiation formulas of the trigonometric functions SUMMARY:
- 10. A. Find the derivative of each of the following functions and simplify the result: ( ) x3sin2xf.1 = ( ) xsin exg.2 = ( ) ( )22 x31cosxh.3 −= ( ) ( )( ) x3cos6 3x3cos2x'f = = ( ) xsin dx d ex'g xsin = ( ) ( )[ ]22 x31cosxh −= x2 1 xcose xsin ⋅⋅= ( ) x2 xcosex x x x2 xcose x'g xsinxsin ⋅ =• ⋅ = ( ) ( )[ ] ( )[ ]( )x6x31sinx31cos2x'h 22 −−−−= ( )[ ] ( )[ ]22 x31sinx31cos2x6 −−= 2sinxcosx2xsinfrom = ( ) ( )2 x312sinx6x'h −=
- 11. 3x4cos3x4sin3y.4 = ( )( )( ) ( )( )( )[ ]233233 x12x4cosx4cosx12x4sinx4sin3'y +−= xsinxcos2xcosfrom 22 −= ( )[ ]32 x42cosx36'y = 32 x8cosx36'y =
- 12. ( ) x 2 x tan2xf.5 −= ( ) 1 2 1 2 x sec2x'f 2 − = ( ) 1 2 x secx'f 2 −= ( ) 2 x tanx'f 2 =
- 13. ( ) x1 x tan 3 logxh.6 − = ( ) ( )( ) ( ) ( ) − −−− ⋅ − ⋅⋅ − = 2 2 3 x1 1x1x1 x1 x secelog x1 x tan 1 x'h ( ) ( )( ) ( ) x1 x cos 1 x1 x sin x1 x cos x1 xx1elog x'h 2 2 3 − ⋅ − −⋅ − +− = ( ) ( ) 2 2 x1 x cos x1 x sin 1 x1 elog x'h 2 3 ⋅ −− ⋅ − = ( ) ( ) ( ) x1 x cos x1 x sin2 1 x1 elog2 x'h 2 3 −− ⋅ − = ( ) ( ) ( ) x1 x2 sin 1 x1 elog2 x'h 2 3 − ⋅ − = ( ) ( ) ( ) −− =⇒ x1 x2 csc x1 elog2 x'h 2 3
- 14. ( ) x2cos x2secy.7 = ( ) ( )x2seclnx2cosyln x2seclnyln sidesbothonlnApply x2cos = = ( ) ( )( )( ) ( )[ ][ ]( )2x2sinx2secln2x2tanx2sec x2sec 1 x2cos'y y 1 ationdifferenticlogarithmiBy −+ =⋅ [ ] x2seclnx2sin2 x2cos x2sin2 x2cos'y y 1 − =⋅ ( )[ ]x2secln1x2sin2 −= ( )[ ] yx2secln1x2sin2'y ⋅−= ( )( )( ) x2cos x2secx2secln1x2sin2'y −=
- 15. ( ) xcot1 xcot2 xh.8 2 + = ( ) ( ) ( )( )[ ] ( )( ) ( )( )[ ] ( )22 222 xcot1 1xcscxcot2xcot21xcsc2xcot1 x'h + −−−+ = ( ) ( ) [ ]xcot1xcot2 xcot1 xcsc2 x'h 22 22 2 −− + = ( ) [ ]1xcot xcsc xcsc2 2 22 2 −= ( ) ( ) ( ) −= − = 1 xsin xcos xsin2 xcsc 1xcot2 x'h 2 2 2 2 2 ( ) ( ) − = xsin xsinxcos xsin2x'h 2 22 2 ( ) x2cos2x'h =
- 16. ( ) ( )1xcscxF.9 3 += ( ) ( ) ( )[ ] ( )1xcsc2 x31xcot1xcsc x'F 3 233 + ++− = ( ) ( ) ( )[ ] ( ) ( )1xcsc2 1xcsc1xcot1xcscx3 x'F 3 3332 + +++ −= ( ) ( ) ( )1xcsc1xcotx 2 3 x'F 332 + +−=
- 17. Find the derivative and simplify the result. ( ) ( ) 3 x4 5sinlnxh.1 = ( ) ( )3 2 xlncosxf.2 = ( ) x4cos2 x4sin xg.3 + = ( ) x2cosx4sin2x2sinxcos2xF.4 −= xcos31 sin y.5 3 − = ( ) ( ) xtan xsinxF. =6 ( )yxtany.7 += ( ) 2 2 x1 x2cot xF.8 + = 0xyxycot.9 =+ EXERCISES: 0ycscxsec.10 22 =+