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Continuity And Differentiability

Question

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Differentiate the following functions w.r.t.x:
$log square root of straight x squared plus straight y squared end root equals tan to the power of negative 1 end exponent straight y over straight x$

Solution

Here space space space log open parentheses straight x squared plus straight y squared close parentheses to the power of 1 half end exponent equals tan to the power of negative 1 end exponent straight y over straight x therefore space 1 half log left parenthesis straight x squared plus straight y squared right parenthesis equals tan to the power of negative 1 end exponent straight y over straight x Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get comma fraction numerator 1 over denominator 2 left parenthesis straight x squared plus straight y squared right parenthesis end fraction open parentheses 2 straight x plus 2 straight y dy over dx close parentheses equals fraction numerator 1 over denominator 1 plus begin display style straight y squared over straight x squared end style end fraction. open parentheses fraction numerator straight x begin display style dy over dx end style minus straight y.1 over denominator straight x squared end fraction close parentheses therefore space fraction numerator straight x plus straight y begin display style dy over dx end style over denominator straight x squared plus straight y squared end fraction equals fraction numerator straight x begin display style dy over dx end style minus straight y over denominator straight x squared plus straight y squared end fraction therefore space space space space straight x plus straight y dy over dx equals straight x dy over dx minus straight y therefore space left parenthesis straight x minus straight y right parenthesis dy over dx equals straight x plus straight y therefore space space space space space space space space space space space space dy over dx equals fraction numerator straight x plus straight y over denominator straight x minus straight y end fraction

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