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Continuity And Differentiability

Question

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$If space straight y equals tan to the power of negative 1 end exponent straight x over straight y comma space then space evaluate space dy over dx.$

Solution

straight y equals tan to the power of negative 1 end exponent straight x over straight y space rightwards double arrow space straight y over straight x equals tan to the power of negative 1 end exponent straight x over straight y space rightwards double arrow space tan straight x over straight y equals straight x over straight y 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 space space space space space space space space space space... left parenthesis 1 right parenthesis Differentiating space both space sides space of space left parenthesis 1 right parenthesis space straight w. straight r. straight t. straight x. space comma open parentheses sec squared straight y over straight x close parentheses. fraction numerator straight x begin display style dy over dx end style minus straight y.1 over denominator straight x squared end fraction equals fraction numerator straight y.1 minus straight x begin display style dy over dx end style over denominator straight y squared end fraction rightwards double arrow space 1 over straight x squared open parentheses 1 plus tan squared straight y over straight x close parentheses open parentheses straight x dy over dx minus straight y close parentheses equals 1 over straight y squared open parentheses straight y minus straight x dy over dx close parentheses rightwards double arrow 1 over straight x squared open parentheses 1 plus straight x squared over straight y squared close parentheses open parentheses straight x dy over dx minus straight y close parentheses plus 1 over straight y squared open parentheses straight x dy over dx minus straight y close parentheses equals 0 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 left square bracket because space of left parenthesis 1 right parenthesis right square bracket space space space space space space open square brackets 1 over straight x squared open parentheses 1 plus straight x squared over straight y squared close parentheses plus 1 over straight y squared open parentheses straight x dy over dx minus straight y close parentheses close square brackets equals 0 rightwards double arrow space straight x dy over dx minus straight y equals 0 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 space space space space space space space space space space space space space space space space space space space space open square brackets because space fraction numerator 1 over denominator straight x squared open parentheses 1 plus straight x squared over straight y squared close parentheses end fraction plus 1 over straight y squared not equal to 0 close square brackets rightwards double arrow space straight x dy over dx equals straight y space rightwards double arrow space dy over dx equals straight x over straight x.

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