Limits and Derivatives

Limits and Derivatives

Question

Find the derivative of the following functions from first principle. 

space space space space space fraction numerator straight x plus 1 over denominator straight x minus 1 end fraction

Answer
space space space space straight f left parenthesis straight x right parenthesis space fraction numerator straight x plus 1 over denominator straight x minus 1 end fraction

space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space therefore space space space space space space space space straight f left parenthesis straight x right parenthesis space equals space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator begin display style fraction numerator straight x plus straight h plus 1 over denominator straight x plus straight h minus 1 end fraction minus fraction numerator straight x plus 1 over denominator straight x minus 1 end fraction end style over denominator straight h end fraction

               space space space space space space space space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of 1 over straight h open square brackets fraction numerator left parenthesis straight x plus 1 plus straight h right parenthesis left parenthesis straight x minus 1 right parenthesis minus left parenthesis straight x minus 1 plus straight h right parenthesis left parenthesis straight x plus 1 right parenthesis over denominator left parenthesis straight x plus straight h minus 1 right parenthesis left parenthesis straight x minus 1 right parenthesis end fraction close square brackets


             space space space space space space space space space space space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of 1 over straight h open square brackets fraction numerator straight x squared minus 1 plus hx minus straight h minus left parenthesis straight x squared minus 1 right parenthesis minus straight h left parenthesis straight x plus 1 right parenthesis over denominator left parenthesis straight x plus straight h minus 1 right parenthesis left parenthesis straight x minus 1 right parenthesis end fraction close square brackets
             space space space space space space space space space space space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of 1 over straight h open square brackets fraction numerator straight x squared minus 1 plus hx minus straight h minus straight x squared plus 1 minus hx minus straight h over denominator left parenthesis straight x plus straight h minus 1 right parenthesis left parenthesis straight x minus 1 right parenthesis end fraction close square brackets


                      <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>


                    

More Chapters from Limits and Derivatives