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Continuity And Differentiability

Question

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Differentiate the following functions w.r.t.x: $tan to the power of negative 1 end exponent left parenthesis sec space straight x plus tan space straight x right parenthesis$

Solution

Let space space straight y equals tan to the power of negative 1 end exponent left parenthesis sec space straight x plus tan space straight x right parenthesis therefore space space space space straight y equals tan to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator cos space straight x end fraction plus fraction numerator sin space straight x over denominator cos space straight x end fraction close parentheses equals tan to the power of negative 1 end exponent open parentheses fraction numerator 1 plus sin space straight x over denominator cos space straight x end fraction close parentheses space space space space space space space space space space equals tan to the power of negative 1 end exponent open square brackets fraction numerator cos squared begin display style straight x over 2 end style plus sin squared begin display style straight x over 2 end style plus 2 sin begin display style straight x over 2 end style cos begin display style straight x over 2 end style over denominator cos 2 begin display style straight x over 2 end style minus sin 2 begin display style straight x over 2 end style end fraction close square brackets space space space space space space space space space space equals tan to the power of negative 1 end exponent open square brackets fraction numerator open parentheses cos straight x over 2 plus sin straight x over 2 close parentheses squared over denominator open parentheses cos straight x over 2 minus sin straight x over 2 close parentheses open parentheses cos straight x over 2 plus sin straight x over 2 close parentheses end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator cos begin display style straight x over 2 end style plus sin begin display style straight x over 2 end style over denominator cos begin display style straight x over 2 end style minus sin begin display style straight x over 2 end style end fraction close square brackets space space space space space space space space space space equals tan to the power of negative 1 end exponent open square brackets fraction numerator begin display style fraction numerator cos begin display style straight x over 2 end style over denominator cos begin display style straight x over 2 end style end fraction end style plus begin display style fraction numerator sin begin display style straight x over 2 end style over denominator cos begin display style straight x over 2 end style end fraction end style over denominator begin display style fraction numerator cos begin display style straight x over 2 end style over denominator cos begin display style straight x over 2 end style end fraction end style minus begin display style fraction numerator sin begin display style straight x over 2 end style over denominator cos begin display style straight x over 2 end style end fraction end style end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets fraction numerator 1 plus tan begin display style straight x over 2 end style over denominator 1 minus tan begin display style straight x over 2 end style end fraction close square brackets space space space space space space space space space space equals tan to the power of negative 1 end exponent open square brackets tan open parentheses straight pi over 2 plus straight x over 2 close parentheses close square brackets equals straight pi over 2 plus straight x over 2 therefore space dy over dx equals 1 half

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