from Mathematics Continuity and Differentiability

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Continuity and Differentiability

Short Answer Type

481.

Prove space that space dy over dx space is space independent space of space straight x comma space when straight y equals cot to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus sin space straight x end root plus square root of 1 minus sin space straight x end root over denominator square root of 1 plus sin space straight x end root minus square root of 1 minus sin space straight x end root end fraction close square brackets comma space open parentheses 0 less than straight x less than straight pi over 2 close parentheses.

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482.

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

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483. $Differentiate space colon space tan to the power of negative 1 end exponent open parentheses fraction numerator square root of 1 plus straight x end root minus square root of 1 minus straight x end root over denominator square root of 1 plus straight x end root plus square root of 1 minus straight x end root end fraction close parentheses space straight w. straight r. straight t. straight x.$

Let space straight y equals space tan to the power of negative 1 end exponent open parentheses fraction numerator square root of 1 plus straight x end root minus square root of 1 minus straight x end root over denominator square root of 1 plus straight x end root plus square root of 1 minus straight x end root end fraction close parentheses Put space straight x equals cos space straight theta therefore space space space straight y equals space tan to the power of negative 1 end exponent open parentheses fraction numerator square root of 1 plus cosθ end root minus square root of 1 minus cosθ end root over denominator square root of 1 plus cosθ end root plus square root of 1 minus cosθ end root end fraction close parentheses equals space tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 2 cos squared straight theta over 2 end root minus square root of 2 sin squared straight theta over 2 end root over denominator square root of 2 cos squared straight theta over 2 end root plus square root of 2 sin squared straight theta over 2 end root end fraction close square brackets space space space space space space space space space equals space tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 2 cos straight theta over 2 minus square root of 2 sin straight theta over 2 over denominator square root of 2 cos straight theta over 2 plus square root of 2 sin straight theta over 2 end fraction close square brackets equals space tan to the power of negative 1 end exponent open square brackets fraction numerator cos straight theta over 2 minus sin straight theta over 2 over denominator cos straight theta over 2 plus sin straight theta over 2 end fraction close square brackets space space space space space space space space space equals space tan to the power of negative 1 end exponent open square brackets fraction numerator begin display style fraction numerator cos begin display style straight theta over 2 end style over denominator cos begin display style straight theta over 2 end style end fraction end style minus begin display style fraction numerator sin begin display style straight theta over 2 end style over denominator cos begin display style straight theta over 2 end style end fraction end style over denominator begin display style fraction numerator cos begin display style straight theta over 2 end style over denominator cos begin display style straight theta over 2 end style end fraction end style plus begin display style fraction numerator sin begin display style straight theta over 2 end style over denominator cos begin display style straight theta over 2 end style end fraction end style end fraction close square brackets equals space tan to the power of negative 1 end exponent open square brackets fraction numerator 1 minus tan begin display style straight theta over 2 end style over denominator 1 plus tan begin display style straight theta 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 4 minus straight theta over 2 close parentheses close square brackets equals straight pi over 4 minus straight theta over 2 therefore space straight y equals straight pi over 4 minus 1 half. cos to the power of negative 1 end exponent straight x dy over dx equals 0 minus 1 half. fraction numerator negative 1 over denominator square root of 1 minus straight x squared end root end fraction

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484. If y = tan^-1

open parentheses fraction numerator straight x to the power of begin display style 1 third end style end exponent plus straight a to the power of begin display style 1 third end style end exponent over denominator 1 minus straight x to the power of begin display style 1 third end style end exponent space straight a to the power of begin display style 1 third end style end exponent end fraction close parentheses

when 0 < a x < 1, prove that

dy over dx

is independent of a.

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485.

straight y equals sin to the power of negative 1 end exponent open curly brackets straight x square root of 1 minus straight x end root minus square root of straight x square root of 1 minus straight x squared end root close curly brackets space semicolon space find space dy over dx comma

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486.

If space straight y equals sin to the power of negative 1 end exponent left parenthesis straight x squared square root of 1 minus straight x squared end root plus straight x square root of 1 minus straight x to the power of 4 end root right parenthesis comma space show space that space dy over dx minus fraction numerator 2 straight x over denominator square root of 1 minus straight x to the power of 4 end root end fraction equals fraction numerator 1 over denominator square root of 1 minus straight x to the power of 4 end root end fraction

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487.

Differentiate space sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction close parentheses space straight w. straight r. straight t. space tan to the power of negative 1 end exponent straight x.

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488.

Differentiate space colon space tan to the power of negative 1 end exponent open parentheses fraction numerator square root of 1 plus straight x squared end root over denominator straight x end fraction close parentheses space straight w. straight r. straight t. space sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction close parentheses.

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489.

Differentiate space tan to the power of negative 1 end exponent open parentheses fraction numerator 3 straight x minus straight x cubed over denominator 1 minus 3 straight x squared end fraction close parentheses space straight w. straight r. straight t. space tan to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 minus straight x squared end fraction close parentheses

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490.

Differentiate space colon space cos 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 space straight w. straight r. straight t. space tan to the power of negative 1 end exponent open parentheses fraction numerator 3 straight x minus straight x cubed over denominator 1 minus 3 straight x squared end fraction close parentheses

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