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Lesson 1 derivative of trigonometric functions

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Lawrence De Vera
Lawrence De Vera

1. The differentiation formulas for the trigonometric functions sin(u), cos(u), tan(u), cot(u), sec(u), and csc(u) are presented in terms of their derivatives du/dx. 2. The derivatives are obtained by applying the basic differentiation formulas for sin(u) and cos(u) along with the quotient, product, and chain rules. 3. Formulas are also provided for deriving the derivatives of inverse trigonometric functions like arcsin, arccos, arctan, etc.

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DIFFERENTIATION OF TRIGONOMETRIC FUNCTIONS
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.
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 =+ =+ =+
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
( ) ( )       = = 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 =
( ) ( )       = = 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 =
( ) ( )       = = 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 =
( ) ( )       = = 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 =
( ) 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:
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 −=
3x4cos3x4sin3y.4 = ( )( )( ) ( )( )( )[ ]233233 x12x4cosx4cosx12x4sinx4sin3'y +−= xsinxcos2xcosfrom 22 −= ( )[ ]32 x42cosx36'y = 32 x8cosx36'y =
( ) x 2 x tan2xf.5 −= ( ) 1 2 1 2 x sec2x'f 2 −            = ( ) 1 2 x secx'f 2 −= ( ) 2 x tanx'f 2 =
( ) 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
( ) 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 −=
( ) 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 =
( ) ( )1xcscxF.9 3 += ( ) ( ) ( )[ ] ( )1xcsc2 x31xcot1xcsc x'F 3 233 + ++− = ( ) ( ) ( )[ ] ( ) ( )1xcsc2 1xcsc1xcot1xcscx3 x'F 3 3332 + +++ −= ( ) ( ) ( )1xcsc1xcotx 2 3 x'F 332 +    +−=
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 =+

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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 =+