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Continuity and Differentiability

Short Answer Type

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

sin left parenthesis tan to the power of negative 1 end exponent space straight e to the power of negative straight x end exponent right parenthesis

86 Views

402. Differentiate the following function w.r.t.x:

cot to the power of negative 1 end exponent square root of straight x

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403. Differentiate the following function w.r.t.x:

sin minus 1 left parenthesis cos space straight x right parenthesis comma space 0 less than straight x less than straight pi over 2

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404. Differentiate the following function w.r.t.x:

cos minus 1 open parentheses fraction numerator straight x over denominator straight x plus 1 end fraction close parentheses

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

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

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406. Differentiate the following function w.r.t.x:

left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis squared

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407. Differentiate the following function w.r.t.x:

straight e to the power of cot to the power of negative 1 end exponent straight x squared end exponent

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408. Differentiate the following function w.r.t.x:

straight x space sec to the power of negative 1 end exponent straight x

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409. Differentiate the following function w.r.t.x:

tan to the power of negative 1 end exponent left parenthesis log space straight x right parenthesis

87 Views

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

cot to the power of negative 1 end exponent square root of fraction numerator 1 plus straight x over denominator 2 end fraction end root

89 Views