Limits and Derivatives

Limits and Derivatives

Question

Find the derivative of the following :



space space space space space space space fraction numerator 1 plus begin display style 1 over straight x end style over denominator 1 minus begin display style 1 over straight x end style end fraction

Answer

Let 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 space space space space space straight y space equals space space fraction numerator 1 plus begin display style 1 over straight x end style over denominator 1 minus begin display style 1 over straight x end style end fraction equals fraction numerator begin display style fraction numerator straight x plus 1 over denominator straight x end fraction end style over denominator begin display style fraction numerator straight x minus 1 over denominator straight x end fraction end style end fraction equals fraction numerator straight x plus 1 over denominator straight x end fraction straight x fraction numerator straight x over denominator straight x minus 1 end fraction equals fraction numerator straight x plus 1 over denominator straight x minus 1 end fraction
Differentiating both sides w.r.t.   x, we  get
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#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)
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#6 {main}</pre>

              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 space space fraction numerator left parenthesis straight x minus 1 right parenthesis left parenthesis 1 plus 0 right parenthesis minus left parenthesis straight x plus 1 right parenthesis left parenthesis 1 minus 0 right parenthesis over denominator left parenthesis straight x minus 1 right parenthesis squared end fraction equals fraction numerator straight x minus 1 minus straight x minus 1 over denominator left parenthesis straight x minus 1 right parenthesis squared end fraction equals fraction numerator negative 2 over denominator left parenthesis straight x minus 1 right parenthesis squared end fraction

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