Question

Differentiate the following functions w.r.t.x: $cot to the power of negative 1 end exponent open parentheses square root of fraction numerator 1 plus cos 3 straight x over denominator 1 plus cos 3 straight x end fraction end root close parentheses$

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Solution

Letv space straight y equals cot to the power of negative 1 end exponent open parentheses square root of fraction numerator 1 plus cos 3 straight x over denominator 1 plus cos 3 straight x end fraction end root close parentheses space space space space space space space space space space equals cot to the power of negative 1 end exponent open parentheses square root of fraction numerator 2 cos squared begin display style fraction numerator 3 straight x over denominator 2 end fraction end style over denominator 2 cos squared begin display style fraction numerator 3 straight x over denominator 2 end fraction end style end fraction end root close parentheses equals cot to the power of negative 1 end exponent open parentheses fraction numerator cos begin display style fraction numerator 3 straight x over denominator 2 end fraction end style over denominator cos space begin display style fraction numerator 3 straight x over denominator 2 end fraction end style end fraction close parentheses equals cot to the power of negative 1 end exponent open parentheses cot fraction numerator 3 straight x over denominator 2 end fraction close parentheses therefore space space space space space straight y equals 3 over 2 straight x space rightwards double arrow space dy over dx equals 3 over 2

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