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Limits and Derivatives

Short Answer Type

41. Find the derivative of the following :

space space space space space space space space fraction numerator 2 over denominator straight x plus 1 end fraction minus fraction numerator straight x squared over denominator 3 straight x minus 1 end fraction

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42. 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

110 Views

43. 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>

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44. Find the derivatives of the following :

(ax+b)ⁿ(cx+ d)ⁿ

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45. Derivative of xⁿ.

Let f(x) = xⁿ $space space space space space space space space space space space space space space space space space space space space space space space space space straight f left parenthesis straight x right parenthesis space equals space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator left parenthesis straight x plus straight h right parenthesis to the power of straight n minus left parenthesis straight x right parenthesis to the power of straight n over denominator straight h end fraction$

$space space space space space space space space space space space space space space space equals space space limit as straight h rightwards arrow 0 of fraction numerator open square brackets straight x open parentheses 1 plus begin display style straight h over straight x end style close parentheses close square brackets to the power of straight n minus straight x to the power of straight n over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator straight x to the power of straight n open parentheses 1 plus begin display style straight h over straight x end style close parentheses to the power of straight n minus straight x to the power of straight n over denominator straight h end fraction$

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

$space space space space space space space space space space equals space space space straight x to the power of straight n space space limit as straight h rightwards arrow 0 of fraction numerator open parentheses 1 plus begin display style nh over straight x end style plus begin display style fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 2 factorial end fraction end style plus begin display style h squared over x squared end style plus.... close parentheses minus 1 over denominator straight h end fraction$

$space space space space space space space space space equals space straight x to the power of straight n limit as straight h rightwards arrow 0 of fraction numerator straight h open square brackets begin display style straight n over straight x end style plus begin display style fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 21 end fraction end style begin display style straight h over straight x squared end style plus..... close square brackets over denominator straight h end fraction$

 $space space space space space space space space space space space space space space space space equals space space straight x to the power of straight n space limit as straight h rightwards arrow 0 of open square brackets straight n over straight x plus fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 2 factorial end fraction space straight h over straight x squared plus.... close square brackets equals straight x to the power of straight n. end exponent straight n over straight x equals nx to the power of straight n minus 1 end exponent$

Hence, $space space space space straight d over dx left parenthesis straight x to the power of straight n right parenthesis space equals space nx to the power of straight n minus 1 end exponent$