Question

Differentiate the following functions w.r.t.x: $tan to the power of negative 1 end exponent open parentheses fraction numerator cos space straight x minus sin space straight x over denominator cos space straight x plus sin space straight x end fraction close parentheses$

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Solution

Let space space space straight y equals tan to the power of negative 1 end exponent open parentheses fraction numerator cos space straight x minus sin space straight x over denominator cos space straight x plus sin space straight x end fraction close parentheses therefore space space space space space space straight y equals tan to the power of negative 1 end exponent open parentheses fraction numerator begin display style fraction numerator cos space straight x over denominator cos space straight x end fraction end style minus begin display style fraction numerator sin space straight x over denominator cos end fraction end style over denominator begin display style fraction numerator cos space straight x over denominator cos space straight x end fraction end style plus begin display style fraction numerator sin space straight x over denominator cos space straight x end fraction end style end fraction close parentheses equals tan to the power of negative 1 end exponent open square brackets fraction numerator 1 minus tan space straight x over denominator 1 plus tan space straight x end fraction close square brackets equals tan to the power of negative 1 end exponent open square brackets tan open parentheses straight pi over 4 minus straight x close parentheses close square brackets equals straight pi over 4 minus straight x therefore space dy over dx equals negative 1

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