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Let space space space space space straight y equals cot to the power of negative 1 end exponent open parentheses fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction close parentheses Put space space space space space straight x equals tan space straight theta therefore space space space space space space space straight y equals cot to the power of negative 1 end exponent open parentheses fraction numerator 1 minus tan space straight theta over denominator 1 plus tan space straight theta end fraction close parentheses equals cot to the power of negative 1 end exponent open square brackets cot open parentheses straight pi over 4 plus straight theta close parentheses close square brackets equals straight pi over 4 plus straight theta equals straight pi over 4 plus tan to the power of negative 1 end exponent straight x therefore space dy over dx equals 0 plus fraction numerator 1 over denominator 1 plus straight x squared end fraction equals fraction numerator 1 over denominator 1 plus straight x squared end fraction

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452.

Find space dy over dx space in space the space following space colon sin to the power of negative 1 end exponent open parentheses 2 straight x square root of 1 minus straight x squared end root close parentheses comma space minus fraction numerator 1 over denominator square root of 2 end fraction less than straight x less than fraction numerator 1 over denominator square root of 2 end fraction

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453.

Differentiate space the space following space straight w. straight r. straight t. straight x colon cos to the power of negative 1 end exponent straight x left parenthesis 2 straight x squared minus 1 right parenthesis comma space straight x greater than 0

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454. Differentiate the following w.r.t.x:

sin to the power of negative 1 end exponent left parenthesis 3 straight x minus 4 straight x cubed right parenthesis comma space minus 1 half less than straight x less than 1 half

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455. Differentiate the following w.r.t.x:

cos to the power of negative 1 end exponent left parenthesis 4 straight x cubed minus 3 straight x right parenthesis comma space 1 half less than straight x less than 1

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456. Differentiate the following w.r.t.x:

sec to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator 2 straight x squared minus 1 end fraction close parentheses comma space 0 less than straight x less than fraction numerator 1 over denominator square root of 2 end fraction

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457. Differentiate the following w.r.t.x:

sec to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator 1 minus 2 straight x squared end fraction close parentheses

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458. Differentiate the following w.r.t.x:

cosec to the power of negative 1 end exponent open parentheses fraction numerator 1 minus straight x squared over denominator 2 straight x end fraction close parentheses

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459. Differentiate the following w.r.t.x:

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

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460.

Find space dy over dx space in space the space following space colon sin to the power of negative 1 end exponent open parentheses fraction numerator 1 minus straight x squared over denominator 1 plus straight x squared end fraction close parentheses comma space 0 less than straight x less than 1

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