from Mathematics Continuity and Differentiability
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    Continuity and Differentiability

    Multiple Choice QuestionsShort Answer Type

    251. If space straight x to the power of straight y equals straight y to the power of straight x comma space prove space that space dy over dx equals fraction numerator begin display style straight y over straight x end style minus log space straight y over denominator begin display style straight x over straight y end style minus log space straight x end fraction
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    252. If space straight y equals straight x to the power of straight y comma space prove space that space dy over dx equals fraction numerator straight y squared over denominator straight x left parenthesis 1 minus straight y space log space straight x right parenthesis end fraction.
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    253. Find space dy over dx when space xy equals straight e to the power of straight x minus straight y end exponent.
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    254. <pre>uncaught exception: <b>file_put_contents(/home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/web/../../../../../../formulas/64/ea/07693cdc340768600ec4c104b639.ini): failed to open stream: Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php at line #12file_put_contents(/home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/web/../../../../../../formulas/64/ea/07693cdc340768600ec4c104b639.ini): failed to open stream: Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php line 12<br />#0 [internal function]: _hx_error_handler(2, 'file_put_conten...', '/home/config_ad...', 12, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php(12): file_put_contents('/home/config_ad...', 'mml=<math xmlns...') #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(48): sys_io_File::saveContent('/home/config_ad...', 'mml=<math xmlns...') #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(112): com_wiris_util_sys_Store->write('mml=<math xmlns...') #4 /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...') #5 /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) #6 /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) #7 {main}</pre>

    Here space space space straight y equals straight a to the power of xy therefore space log space straight y equals log space straight a to the power of xy space space space space space space space space space space space space space space space space space space rightwards double arrow space log space straight y equals left square bracket xy right square bracket log space straight a Differentiate space straight w. straight r. straight t. straight x comma space space space space space space 1 over straight y dy over dx equals open square brackets straight x. dy over dx plus straight y.1 close square brackets log space straight a therefore space space 1 over straight y dy over dx equals straight x space log space straight a dy over dx plus straight y space log space straight a space space space space space rightwards double arrow space space space space space space open parentheses 1 over straight x minus straight x space log space straight a close parentheses dy over dx equals straight y space log space straight a rightwards double arrow space space space open parentheses fraction numerator 1 minus xy space log space straight a over denominator straight y end fraction close parentheses dy over dx equals straight y space log space straight a space space space space space rightwards double arrow space space space space space space dy over dx equals fraction numerator straight y squared log space straight a over denominator 1 minus xy space log space straight a end fraction
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    255. If space straight y equals square root of fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction end root comma space prove space that space left parenthesis 1 minus straight x squared right parenthesis dy over dx plus straight y equals 0
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    256. If space straight f left parenthesis straight x right parenthesis equals open parentheses fraction numerator 3 plus straight x over denominator 1 plus straight x end fraction close parentheses to the power of 2 plus 3 straight x end exponent comma space find space straight f apostrophe left parenthesis 0 right parenthesis
    83 Views
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    257. Differentiate space square root of fraction numerator left parenthesis straight x minus 3 right parenthesis left parenthesis straight x squared plus 4 right parenthesis over denominator 3 straight x squared plus 4 straight x plus 5 end fraction end root straight w. straight r. straight t. straight x
    79 Views
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    258. Differentiate space square root of fraction numerator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x minus 2 right parenthesis over denominator left parenthesis straight x minus 3 right parenthesis left parenthesis straight x minus 4 right parenthesis left parenthesis straight x minus 5 right parenthesis end fraction end root straight w. straight r. straight t. straight x
    91 Views
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    259. Differentiate : (x + 3)2 (x + 4)3 (x + 5)4.w.r.t.x.
    81 Views
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    260.

    Find the derivative of the function given by

    f(x)=(1 + x)(1 + x2)(1 + x4)(1 + x8)and hence find f'(1).

     
    73 Views
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