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431. Differentiate the following functions w.r.t.x:

tan to the power of negative 1 end exponent open parentheses square root of fraction numerator 1 minus cos space 2 straight x over denominator 1 plus cos space 2 straight x end fraction end root close parentheses

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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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433. Differentiate the following functions w.r.t.x:

cos to the power of negative 1 end exponent open parentheses square root of fraction numerator 1 plus cos space straight x over denominator 2 end fraction end root close parentheses

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434. 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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435. Differentiate the following functions w.r.t.x:

tan to the power of negative 1 end exponent open parentheses square root of fraction numerator 1 minus sin space straight x over denominator 1 plus sin space straight x end fraction end root close parentheses

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436. 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 over denominator 1 plus sin space straight x end fraction close parentheses

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437. 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

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438. Differentiate the following functions w.r.t.x:

tan to the power of negative 1 end exponent open parentheses fraction numerator 1 minus cos space straight x over denominator sin space straight x end fraction close parentheses

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439. Differentiate the following functions w.r.t.x:

tan to the power of negative 1 end exponent open parentheses fraction numerator sin space straight x over denominator 1 plus cos space straight x end fraction close parentheses

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440. Differentiate the following functions w.r.t.x:

cot to the power of negative 1 end exponent open parentheses fraction numerator 1 plus cos space straight x over denominator sin space straight x end fraction close parentheses

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