Exercise:
nprvmulticols abclist abc displaystyle x texte^x ddx abc displaystyle x sinx ddx abc displaystyle x cosx ddx abc displaystyle x lnx ddx abc displaystyle x^ texte^x ddx abc displaystyle x^ cosx ddx abc displaystyle x^ sinx ddx abc displaystyle x^ texte^x ddx abc displaystyle lnx ddx abc displaystyle x texte^x ddx abc displaystyle x lnx + ddx abc displaystyle x arctanx ddx abc displaystyle lnx texte^x ddx abc displaystyle x^ lnx ddx abc displaystyle lnx^ ddx abc displaystyle x cosx ddx abc displaystyle x^ texte^-x ddx abc displaystyle x texte^x ddx abc displaystyle lnx cosx ddx abc displaystyle x arcsinx ddx abc displaystyle x lnax ddx abc displaystyle x^ lnbx ddx abc displaystyle x tan^-x^ ddx abc displaystyle lnx lnx + ddx abclist nprvmulticols

Solution:
abclist abc underbracex_uunderbracemathrme^x_v'ddx underbracex_uunderbracemathrme^x_v - underbrace_u'underbracemathrme^x_vddx x-mathrme^x + C abc underbracex_uunderbracesin x_v'ddx underbracex_uunderbrace-cos x_v - underbrace_u'underbrace-cos x_vddx -xcos x + sin x + C abc underbracex_uunderbracecos x_v'ddx underbracex_uunderbracesin x_v - underbrace_u'underbracesin x_vddx xsin x + cos x + C abc underbracex_v'underbraceln x_uddx underbracetfracx^_vunderbraceln x_u - underbracetfracx_u'underbracetfracx^_vddx fracx^ln x - fracx^ + C abc underbracex^_uunderbracemathrme^x_v'ddx underbracex^_uunderbracemathrme^x_v - underbracex_u'underbracemathrme^x_vddx x^-x+mathrme^x + C abc underbracex^_uunderbracecos x_v'ddx underbracex^_uunderbracesin x_v - underbracex_u'underbracesin x_vddx x^sin x + xcos x - sin x + C abc underbracex^_uunderbracesin x_v'ddx underbracex^_uunderbrace-cos x_v - underbracex_u'underbrace-cos x_vddx -x^cos x + xsin x + cos x + C abc underbracex^_uunderbracemathrme^x_v'ddx underbracex^_uunderbracemathrme^x_v - underbracex^_u'underbracemathrme^x_vddx x^-x^+x-mathrme^x + C abc underbrace_v'underbraceln x_uddx underbracex_vunderbraceln x_u - underbracetfracx_u'underbracex_vddx xln x - x + C abc underbracex_uunderbracemathrme^x_v'ddx underbracex_uunderbracetfracmathrme^x_v - underbrace_u'underbracetfracmathrme^x_vddx fracx-mathrme^x + C abc underbracex_v'underbracelnx+_uddx underbracetfracx^_vunderbracelnx+_u - underbracetfracx+_u'underbracetfracx^_vddx fracx^lnx+ - fracx^ + fracx - fraclnx+ + C abc underbracex_v'underbracearctan x_uddx underbracetfracx^_vunderbracearctan x_u - underbracetfrac+x^_u'underbracetfracx^_vddx tfrac!bigx^+arctan x - xbig + C abc underbracemathrme^x_uunderbraceln x_v'ddx underbracemathrme^x_uunderbracexln x - x_v - underbracemathrme^x_u'underbraceln x - _v'ddx tfracmathrme^xbigxln x - x + big + C abc underbracex^_v'underbraceln x_uddx underbracetfracx^_vunderbraceln x_u - underbracetfracx_u'underbracetfracx^_vddx fracx^ln x - fracx^ + C abc underbrace_v'underbraceln x^_uddx underbracex_vunderbraceln x^_u - underbracetfracln xx_u'underbracex_vddx xln x^ - xln x + x + C abc underbracex_uunderbracecosx_v'ddx underbracex_uunderbracetfracsinx_v - underbrace_u'underbracetfracsinx_vddx tfracxsinx + tfraccosx + C abc underbracex^_uunderbracemathrme^-x_v'ddx underbracex^_uunderbrace-mathrme^-x_v - underbracex_u'underbrace-mathrme^-x_vddx -x^+x+mathrme^-x + C abc underbracex_uunderbracemathrme^x_v'ddx underbracex_uunderbracetfracmathrme^x_v - underbrace_u'underbracetfracmathrme^x_vddx tfracx-mathrme^x + C abc underbraceln x_uunderbracecos x_v'ddx underbraceln x_uunderbracesin x_v - underbracetfracx_u'underbracesin x_vddx ln xsin x - mathrmSix + C abc underbracex_v'underbracearcsin x_uddx underbracetfracx^_vunderbracearcsin x_u - underbracetfracsqrt-x^_u'underbracetfracx^_vddx frac!Bigxsqrt-x^ + x^-arcsin xBig + C abc underbracex_v'underbracelnax_uddx underbracetfracx^_vunderbracelnax_u - underbracetfracx_u'underbracetfracx^_vddx fracx^lnax - fracx^ + C abc underbracex^_v'underbracelnbx_uddx underbracetfracx^_vunderbracelnbx_u - underbracetfracx_u'underbracetfracx^_vddx fracx^lnbx - fracx^ + C abc underbracex_v'underbracearctanx^_uddx underbracetfracx^_vunderbracearctanx^_u - underbracetfracx+x^_u'underbracetfracx^_vddx fracx^arctanx^ - fracln+x^ + C abc underbracelnx+_v'underbraceln x_uddx underbracex+lnx+-x_vunderbraceln x_u - underbracetfracx_u'underbracex+lnx+-x_vddx ln xbigx+lnx+-xbig - bigx+lnx+-xbig - operatornameLi_-x + x + C abclist