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Limits And Derivatives

Question

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Find the derivatives of the following :

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

Solution

$space space space space space space space space space space space straight y space equals space fraction numerator px to the power of 2 space end exponent plus qx plus straight r over denominator ax plus straight b end fraction$
Differentiating both sides w.r.t. x, we get

space space space space space space space space space space space space space space space space space space dy over dx equals straight d over dx open parentheses fraction numerator px squared plus qx plus straight r over denominator ax plus straight b end fraction close parentheses

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 equals space fraction numerator left parenthesis ax plus straight b right parenthesis begin display style straight d over dx end style left parenthesis px squared plus qx plus straight r right parenthesis space minus left parenthesis px squared plus qx plus straight r right parenthesis begin display style straight d over dx end style left parenthesis ax plus straight b right parenthesis space over denominator left parenthesis ax plus straight b right parenthesis squared 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 space equals space space fraction numerator left parenthesis ax plus straight b right parenthesis left parenthesis 2px plus straight q right parenthesis minus left parenthesis px squared plus qx plus straight r right parenthesis left parenthesis straight a plus 0 right parenthesis over denominator left parenthesis ax plus straight b right parenthesis squared 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 equals space space space space space fraction numerator 2 ap space straight x squared aqx plus 2 bpx plus bq minus apx squared minus aqx minus ar over denominator left parenthesis ax plus straight b right parenthesis squared end fraction

<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>

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