MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C97CE5.50157BB0" This document is a Web archive file. If you are seeing this message, this means your browser or editor doesn't support Web archive files. For more information on the Web archive format, go to http://officeupdate.microsoft.com/office/webarchive.htm ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions

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------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/master24.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/preview.wmf Content-Transfer-Encoding: base64 Content-Type: image/x-wmf AQAJAAADzwEAAAUAKgAAAAAABAAAAAMBCAAFAAAACwIAAAAABQAAAAwC7wLpAwMAAAAeAAcAAAD8 AgAA////AAAABAAAAC0BAAAIAAAA+gIFAAAAAAD///8ABAAAAC0BAQAMAAAAQAkhAPAAAAAAAAAA 7wLpA/////8IAAAA+gIAAAAAAAAAAAAABAAAAC0BAgAEAAAALQEAAAQAAAAnAf//HAAAAPsC3/8A AAAAAAC8AgAAAAAAQAAiQXJpYWwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEAAAALQEDAAQA AAAuARgABAAAAAIBAQAFAAAACQIAAAACHwAAADIKIgDGABAAAABEZXJpdmF0aXZlcyBhbmQgGAAS AA0ACQATABIACwAJABQAEgASAAoAEgAVABQACQAEAAAALgEAABwAAAD7AhAABwAAAAAAvAIAAAAA AQICIlN5c3RlbQAAAAAAABgAAABoaRMAAQAAAOQEAAAAAAAABAAAAC0BBAAEAAAA8AEDABwAAAD7 At//AAAAAAAAvAIAAAAAAEAAIkFyaWFsAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABAAAAC0B AwAEAAAALgEYAAQAAAACAQEABQAAAAkCAAAAAhkAAAAyCiIAxQEMAAAASW50ZWdyYWxzIG9mCQAU AAwAEgAUAA4AEgAKABIACQAUAAsABAAAAC4BAAAEAAAALQEEAAQAAADwAQMAHAAAAPsC3/8AAAAA AAC8AgAAAAAAQAAiQXJpYWwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEAAAALQEDAAQAAAAu ARgABAAAAAIBAQAFAAAACQIAAAACGQAAADIKIgCCAgwAAAB0aGUgSW52ZXJzZSALABQAEwAJAAkA FAATABIADgASABMACgAEAAAALgEAAAQAAAAtAQQABAAAAPABAwAcAAAA+wLf/wAAAAAAALwCAAAA AABAACJBcmlhbAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAQAAAAtAQMABAAAAC4BGAAEAAAA AgEBAAUAAAAJAgAAAAIqAAAAMgpKADgBFwAAAFRyaWdvbm9tZXRyaWMgRnVuY3Rpb25zABUADQAJ ABQAFAAUABQAHgASAAsADQAKABIACQAVABQAFQASAAsACQAVABQAEgAEAAAALgEAAAQAAAAtAQQA BAAAAPABAwADAAAAAAA= ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0001.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
Derivatives and Integrals of the Inverse Trigonometric Functions
------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0008.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
Suppo= se f and g is a pair of inverse functions, so
(f o g)(x) =3D x.
Derivatives and Integrals of the Inverse Trigonometric Functions
------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0009.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
Suppo= se f and g is a pair of inverse functions, so
(f o g)(x) =3D x.
Differentiate both sides with respect to x and uses the chain rule:
[(f <= /span>o g)(x)]'  <= /span>=3D x'
Derivatives and Integrals of the Inverse Trigonometric Functions
------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0010.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
Ó=  <= /span>Frank Ma 2006
Suppo= se f and g is a pair of inverse functions, so
(f o g)(x) =3D x.
Differentiate both sides with respect to x and uses the chain rule:
[(f <= /span>o g)(x)]'  <= /span>=3D x'
  f '(g(x)) * g'(x)  =3D 1 
Derivatives and Integrals of the Inverse Trigonometric Functions
------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0011.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
Ó=  <= /span>Frank Ma 2006
Suppo= se f and g is a pair of inverse functions, so
(f o g)(x) =3D x.
Differentiate both sides with respect to x and uses the chain rule:
[(f <= /span>o g)(x)]'  <= /span>=3D x'
  f '(g(x)) * g'(x)  =3D 1 
or     
g'(x)=   =3D   
1
f '(<= /span>g(x))
Derivatives and Integrals of the Inverse Trigonometric Functions
------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0011_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA7 AAEAgAAAAAAAAAIGjI+py+1YADt= ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0012.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
Ó=  <= /span>Frank Ma 2006
Suppo= se f and g is a pair of inverse functions, so
(f o g)(x) =3D x.
Differentiate both sides with respect to x and uses the chain rule:
[(f <= /span>o g)(x)]'  <= /span>=3D x'
  f '(g(x)) * g'(x)  =3D 1 
or     
g'(x)=   =3D   
1
f '(<= /span>g(x))
Set f = =3D sin(x) and g =3D arcsin(x)
Derivatives and Integrals of the Inverse Trigonometric Functions
------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0012_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA7 AAEAgAAAAAAAAAIGjI+py+1YADt= ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0013.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
Ó=  <= /span>Frank Ma 2006
Suppo= se f and g is a pair of inverse functions, so
(f o g)(x) =3D x.
Differentiate both sides with respect to x and uses the chain rule:
[(f <= /span>o g)(x)]'  <= /span>=3D x'
  f '(g(x)) * g'(x)  =3D 1 
or     
g'(x)=   =3D   
1
f '(<= /span>g(x))
Set f = =3D sin(x) and g =3D arcsin(x) we obtain
[arcs= in(x)]'  =3D   
1
 dsin(y)
Derivatives and Integrals of the Inverse Trigonometric Functions
 dy
y= =3Darcsin(x)
------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0013_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA7 AAEAgAAAAAAAAAIGjI+py+1YADt= ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0013_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhaQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBn AAEAgAAAAAAAAAIIjI+py+0PIywAOy== ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0013_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAy AAEAgAAAAAAAAAIGjI+py50FADs= ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0013_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAiAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAgAC AB8AgAAAAAAAAAIGjI+py+1bADt= ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0014.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
Ó=  <= /span>Frank Ma 2006
Suppo= se f and g is a pair of inverse functions, so
(f o g)(x) =3D x.
Differentiate both sides with respect to x and uses the chain rule:
[(f <= /span>o g)(x)]'  <= /span>=3D x'
  f '(g(x)) * g'(x)  =3D 1 
or     
g'(x)=   =3D   
1
f '(<= /span>g(x))
Set f = =3D sin(x) and g =3D arcsin(x) we obtain
[arcs= in(x)]'  =3D   
1
 dsin(y)
1
cos(<= /span>arcsin(x))<= /span>
 =3D   
Derivatives and Integrals of the Inverse Trigonometric Functions
 dy
y= =3Darcsin(x)
------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0014_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA7 AAEAgAAAAAAAAAIGjI+py+1YADt= ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0014_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhaQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBn AAEAgAAAAAAAAAIIjI+py+0PIywAOy== ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0014_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhaQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBn AAEAgAAAAAAAAAIIjI+py+0PIywAOy== ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0014_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAy AAEAgAAAAAAAAAIGjI+py50FADs= ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0014_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgAiAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAgAC AB8AgAAAAAAAAAIGjI+py+1bADt= ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0002.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
[arcsi= n(x)]'   
1
cos(<= /span>arcsin(x))<= /span>
 =3D   
Derivatives and Integrals of the Inverse Trigonometric Functions
------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0002_image012.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhaAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABm AAEAgAAAAAAAAAIIjI+py+0P4ysAOy== ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0015.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
θ=3Darcsin(x)
x
1
[arcsi= n(x)]'   
1
cos(<= /span>arcsin(x))<= /span>
 =3D   
Derivatives and Integrals of the Inverse Trigonometric Functions
------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0015_image013.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhkwBFAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAQCP AEQAgAAAAAAAAAL/hI+py+0Po5xUhYqz3ryH34XiSEYgcJXqylankbbyLL9HTOf6ZiP4Dgw2eomf 8Aj8GC3I5o44dEpbUMdyis1UH9esF7IFf8eSsImMZpjL6TZqPem6m3DX3FvHyO+6vJZPtyciCKji x0FYOJg4wqiodTj4yBJJ4jgJdrmiiblQudmJyEkVCkk2OvlJg1qomsM6p3QHm+YaRDtmK4SbpXvE K+WLBDxM/BsqHJyaPGU8w9wLCI036/xl3Yh9Xaude9p9ijc9izUuHQxOXlxq+ZveevuuKG/ODi7L XlMjX2pdn1/EED+AwP4B9LTo4DFRCofxGNjQhx6DEa1QoFiRSxmILRkRcuHY0aMakCFFFiFZ0iQM lClF4mu5DSPMiUpq2ryJM6fOnTx7+vwJNKjOAgA7 ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0015_image014.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhaAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABm AAEAgAAAAAAAAAIIjI+py+0P4ysAOy== ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0016.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
θ=3Darcsin(x)
x
1
Ö1 – x2=
[arcsi= n(x)]'   
1
cos(<= /span>arcsin(x))<= /span>
 =3D   
Derivatives and Integrals of the Inverse Trigonometric Functions
------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0016_image015.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhkwBFAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAQCP AEQAgAAAAAAAAAL/hI+py+0Po5xUhYqz3ryH34XiSEYgcJXqylankbbyLL9HTOf6ZiP4Dgw2eomf 8Aj8GC3I5o44dEpbUMdyis1UH9esF7IFf8eSsImMZpjL6TZqPem6m3DX3FvHyO+6vJZPtyciCKji x0FYOJg4wqiodTj4yBJJ4jgJdrmiiblQudmJyEkVCkk2OvlJg1qomsM6p3QHm+YaRDtmK4SbpXvE K+WLBDxM/BsqHJyaPGU8w9wLCI036/xl3Yh9Xaude9p9ijc9izUuHQxOXlxq+ZveevuuKG/ODi7L XlMjX2pdn1/EED+AwP4B9LTo4DFRCofxGNjQhx6DEa1QoFiRSxmILRkRcuHY0aMakCFFFiFZ0iQM lClF4mu5DSPMiUpq2ryJM6fOnTx7+vwJNKjOAgA7 ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0016_image016.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAp AAEAgAAAAP8AAAIFjI+py1kAO1== ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0016_image017.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhaAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABm AAEAgAAAAAAAAAIIjI+py+0P4ysAOy== ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0017.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
θ=3Darcsin(x)
x
1
Ö1 – x2=
[arcsi= n(x)]'   
1
cos(<= /span>arcsin(x))<= /span>
 =3D   
1
Ö1 – x2=
[arcsi= n(x)]'  =3D   
Derivatives and Integrals of the Inverse Trigonometric Functions
------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0017_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhkwBFAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAQCP AEQAgAAAAAAAAAL/hI+py+0Po5xUhYqz3ryH34XiSEYgcJXqylankbbyLL9HTOf6ZiP4Dgw2eomf 8Aj8GC3I5o44dEpbUMdyis1UH9esF7IFf8eSsImMZpjL6TZqPem6m3DX3FvHyO+6vJZPtyciCKji x0FYOJg4wqiodTj4yBJJ4jgJdrmiiblQudmJyEkVCkk2OvlJg1qomsM6p3QHm+YaRDtmK4SbpXvE K+WLBDxM/BsqHJyaPGU8w9wLCI036/xl3Yh9Xaude9p9ijc9izUuHQxOXlxq+ZveevuuKG/ODi7L XlMjX2pdn1/EED+AwP4B9LTo4DFRCofxGNjQhx6DEa1QoFiRSxmILRkRcuHY0aMakCFFFiFZ0iQM lClF4mu5DSPMiUpq2ryJM6fOnTx7+vwJNKjOAgA7 ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0017_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAp AAEAgAAAAP8AAAIFjI+py1kAO1== ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0017_image020.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhaAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABm AAEAgAAAAAAAAAIIjI+py+0P4ysAOy== ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0017_image021.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhaQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBn AAEAgAAAAAAAAAIIjI+py+0PIywAOy== ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0017_image022.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAp AAEAgAAAAP8AAAIFjI+py1kAO1== ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0018.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
θ=3Darcsin(x)
x
1
Ö1 – x2=
[arcsi= n(x)]'   
1
cos(<= /span>arcsin(x))<= /span>
 =3D   
1
Ö1 – x2=
[arcsi= n(x)]'  =3D   
we use the same technique to obtain the derivatives of the other inverse trig. functions.
Derivatives and Integrals of the Inverse Trigonometric Functions
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θ=3Darcsin(x)
x
1
Ö1 – x2=
[arcsi= n(x)]'   
1
cos(<= /span>arcsin(x))<= /span>
 =3D   
1
Ö1 – x2=
[arcsi= n(x)]'  =3D   
we use the same technique to obtain the derivatives of the other inverse trig. functions. They are listed below in the general form where &= #13;
u =3D u(x) .
Derivatives and Integrals of the Inverse Trigonometric Functions
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Derivatives and Integrals of the Inverse Trigonometric Functions
u'
Ö1 – u2
[arcsi= n(u)]'  =3D   
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Derivatives and Integrals of the Inverse Trigonometric Functions
u'
Ö1 – u2
[arcsi= n(u)]'  =3D   
- u'
Ö1 – u2
[arcco= s(u)]'  =3D   
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Derivatives and Integrals of the Inverse Trigonometric Functions
u'
Ö1 – u2
[arcsi= n(u)]'  =3D   
- u'
Ö1 – u2
[arcco= s(u)]'  =3D   
u'
 1 + u2
[arcta= n(u)]'  =3D   
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Derivatives and Integrals of the Inverse Trigonometric Functions
u'
Ö1 – u2
[arcsi= n(u)]'  =3D   
- u'
Ö1 – u2
[arcco= s(u)]'  =3D   
u'
 1 + u2
[arcta= n(u)]'  =3D   
u'
[arcse= c(u)]'  =3D   
|u|Öu2 – 1
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Derivatives and Integrals of the Inverse Trigonometric Functions
u'
Ö1 – u2
[arcsi= n(u)]'  =3D   
- u'
Ö1 – u2
[arcco= s(u)]'  =3D   
u'
 1 + u2
[arcta= n(u)]'  =3D   
u'
[arcse= c(u)]'  =3D   
d=
dx
sin-= 1(u) =3D &nb= sp; 
1
Ö1 – u2
du
dx
d
dx
cos-= 1(u) =3D &nb= sp; 
-1
Ö1 – u2
du
dx
d
dx
tan-= 1(u) =3D &nb= sp; 
1
1 + u= 2=
du
dx
d
dx
 sec-1(u) =3D &nb= sp; 
1
|u|Öu2 – 1
du
dx
|u|Öu2 – 1
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file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0022_image071.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAg AAEAgAAAAAAAAAIFjI+piwUAOw== ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0022_image072.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQABAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABE AAEAgAAAAAAAAAIHjI+py+1vCgA7 ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0023.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
Derivatives and Integrals of the Inverse Trigonometric Functions
u'
Ö1 – u2
[arcsi= n(u)]'  =3D   
- u'
Ö1 – u2
[arcco= s(u)]'  =3D   
u'
 1 + u2
[arcta= n(u)]'  =3D   
u'
[arcse= c(u)]'  =3D   
d=
dx
sin-= 1(u) =3D &nb= sp; 
1
Ö1 – u2
du
dx
d
dx
cos-= 1(u) =3D &nb= sp; 
-1
Ö1 – u2
du
dx
d
dx
tan-= 1(u) =3D &nb= sp; 
1
1 + u= 2=
du
dx
d
dx
 sec-1(u) =3D &nb= sp; 
1
|u|Öu2 – 1
du
dx
Example: Find the derivative of tan-1(x3)  
|u|Öu2 – 1
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file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0023_image093.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAg AAEAgAAAAAAAAAIFjI+piwUAOw== ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0023_image094.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQABAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABE AAEAgAAAAAAAAAIHjI+py+1vCgA7 ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0024.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
Derivatives and Integrals of the Inverse Trigonometric Functions
u'
Ö1 – u2
[arcsi= n(u)]'  =3D   
- u'
Ö1 – u2
[arcco= s(u)]'  =3D   
u'
 1 + u2
[arcta= n(u)]'  =3D   
u'
[arcse= c(u)]'  =3D   
d=
dx
sin-= 1(u) =3D &nb= sp; 
1
Ö1 – u2
du
dx
d
dx
cos-= 1(u) =3D &nb= sp; 
-1
Ö1 – u2
du
dx
d
dx
tan-= 1(u) =3D &nb= sp; 
1
1 + u= 2=
du
dx
d
dx
 sec-1(u) =3D &nb= sp; 
1
|u|Öu2 – 1
du
dx
Example: Find the derivative of tan-1(x3)  
u =3D x3,   
|u|Öu2 – 1
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file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0024_image115.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAg AAEAgAAAAAAAAAIFjI+piwUAOw== ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0024_image116.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQABAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABE AAEAgAAAAAAAAAIHjI+py+1vCgA7 ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0025.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
Derivatives and Integrals of the Inverse Trigonometric Functions
u'
Ö1 – u2
[arcsi= n(u)]'  =3D   
- u'
Ö1 – u2
[arcco= s(u)]'  =3D   
u'
 1 + u2
[arcta= n(u)]'  =3D   
u'
[arcse= c(u)]'  =3D   
d=
dx
sin-= 1(u) =3D &nb= sp; 
1
Ö1 – u2
du
dx
d
dx
cos-= 1(u) =3D &nb= sp; 
-1
Ö1 – u2
du
dx
d
dx
tan-= 1(u) =3D &nb= sp; 
1
1 + u= 2=
du
dx
d
dx
 sec-1(u) =3D &nb= sp; 
1
|u|Öu2 – 1
du
dx
Example: Find the derivative of tan-1(x3)  
u =3D x3, so [tan-1(x3)]' =3D  
1
1 + (x3)2
dx3
dx
|u|Öu2 – 1
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file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0025_image140.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRQABAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABE AAEAgAAAAAAAAAIHjI+py+1vCgA7 ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0026.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
Derivatives and Integrals of the Inverse Trigonometric Functions
u'
Ö1 – u2
[arcsi= n(u)]'  =3D   
- u'
Ö1 – u2
[arcco= s(u)]'  =3D   
u'
 1 + u2
[arcta= n(u)]'  =3D   
u'
|u|Öu2 – 1
[arcse= c(u)]'  =3D   
d=
dx
sin-= 1(u) =3D &nb= sp; 
1
Ö1 – u2
du
dx
d
dx
cos-= 1(u) =3D &nb= sp; 
-1
Ö1 – u2
du
dx
d
dx
tan-= 1(u) =3D &nb= sp; 
1
1 + u= 2=
du
dx
d
dx
 sec-1(u) =3D &nb= sp; 
1
|u|Öu2 – 1
du
dx
Example: Find the derivative of tan-1(x3)  
u =3D x3, so [tan-1(x3)]' =3D  
1
1 + (x3)2
dx3
dx
=3D   
3x2
1 + x6
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file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0026_image164.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAg AAEAgAAAAAAAAAIFjI+piwUAOw== ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0026_image165.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhSQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABH AAEAgAAAAAAAAAIHjI+py+2fCgA7 ------=_NextPart_01C97CE5.50157BB0 Content-Location: file:///C:/4D764A33/15DerivativesandIntegralsofInverseTrigonometricFunctions_files/slide0061.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The Inverse Trigonometric Functions
Derivatives and Integrals of the Inverse Trigonometric Functions
u'
[arcs= ec(u)]'  =3D   
[cos-= 1(u)]' =3D   
-u'
Ö1 – u2
Example: a. Find the derivative of cos-1(ex  )  
2
|u|Öu2 – 1
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Derivatives and Integrals of the Inverse Trigonometric Functions
u'
[arcs= ec(u)]'  =3D   
[cos-= 1(u)]' =3D   
-u'
Ö1 – u2
Example: a. Find the derivative of cos-1(ex  )  
u =3D ex,  
2
2
|u|Öu2 – 1
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