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Limits and Derivatives

Short Answer Type

51.

Find the derivative of $space space space straight y space equals space 10 to the power of 10 plus 10 over straight x plus straight x over 10 plus 10 straight x plus straight x to the power of 10$

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52.

Find the derivative of (2x+3)(5x²-7x+1)

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53. Differentiate the following functions w.r.t. x

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 fraction numerator straight x minus 1 over denominator 3 straight x cubed plus 2 end fraction comma space straight x space not equal to space cube root of fraction numerator negative 2 over denominator 3 end fraction end root

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54.

Solve $space space space space space space space space space space limit as straight z rightwards arrow 1 of fraction numerator straight z to the power of begin display style 1 third end style end exponent minus 1 over denominator straight z to the power of begin display style 1 over 6 end style end exponent minus 1 end fraction$

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55.

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

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

It is of $0 over 0$ form.

∴    $space space space space space space space space space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow 1 of fraction numerator left parenthesis straight x plus straight x squared plus straight x cubed plus.... plus straight x to the power of straight n right parenthesis minus straight n over denominator straight x minus 1 end fraction equals limit as straight x rightwards arrow 1 of fraction numerator left parenthesis straight x minus 1 right parenthesis plus left parenthesis straight x squared minus 1 right parenthesis plus left parenthesis straight x cubed minus 1 right parenthesis plus... plus left parenthesis straight x to the power of straight n minus 1 right parenthesis over denominator straight x minus 1 end fraction$

   $space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space limit as x rightwards arrow 1 of open square brackets 1 plus fraction numerator x squared minus 1 over denominator x minus 1 end fraction plus fraction numerator x cubed minus 1 over denominator x minus 1 end fraction plus.... plus fraction numerator x to the power of n minus 1 over denominator x minus 1 end fraction close square brackets$

   $space space space space space space space space space space space space space space space space space space space space space equals space limit as straight x rightwards arrow 1 of 1 plus limit as straight x rightwards arrow of fraction numerator straight x squared minus 1 over denominator straight x minus 1 end fraction plus limit as straight x rightwards arrow 1 of fraction numerator straight x cubed minus 1 over denominator straight x minus 1 end fraction plus...... plus limit as straight x rightwards arrow 1 of fraction numerator straight x to the power of straight n minus 1 over denominator straight x minus 1 end fraction$

= 1 + 2(1)²+ 3(1)²+ ......+ n(1)^n-1= 1 + 2 + 3 + ..... +n


   $space space space space space space space space space equals space sum subscript straight n fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction$