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Continuity and Differentiability

Short Answer Type

461.

Find space dy over dx in space the space following space colon cos 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 comma negative 1 less than straight x less than 1

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462.

Differentiate space the space following space functions space by space substitutions space method space colon cos 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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463. Differentiate the following functions by substitutions method :

sin to the power of negative 1 end exponent square root of fraction numerator 1 plus straight x over denominator 2 end fraction end root comma negative 1 less than x less than 1

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464. Differentiate the following functions by substitutions method :

cot to the power of negative 1 end exponent open parentheses fraction numerator square root of 1 plus straight x squared end root minus 1 over denominator straight x end fraction close parentheses comma space straight x not equal to 0

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465. Differentiate the following functions by substitutions method :

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

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466. Differentiate the following functions by substitutions method :

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

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

tan to the power of negative 1 end exponent left parenthesis square root of 1 plus straight x squared end root minus straight x right parenthesis

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

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

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

cot to the power of negative 1 end exponent open parentheses square root of 1 plus straight x squared end root minus straight x close parentheses

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470. Differentiate the following w.r.t.x: $cot to the power of negative 1 end exponent open parentheses square root of 1 plus straight x squared end root plus straight x close parentheses$

Let space space straight y equals cot to the power of negative 1 end exponent open parentheses square root of 1 plus straight x squared end root plus straight x close parentheses semicolon space Put space straight x equals tan space straight theta therefore space space space straight y equals cot to the power of negative 1 end exponent left square bracket square root of 1 plus tan squared straight theta end root plus tanθ right square bracket equals cot to the power of negative 1 end exponent left square bracket sec space straight theta plus an space straight theta right square bracket space space space space space space space space space equals cot to the power of negative 1 end exponent open square brackets fraction numerator 1 over denominator cos space straight theta end fraction plus fraction numerator sin space straight theta over denominator cos space straight theta end fraction close square brackets equals cot to the power of negative 1 end exponent open square brackets fraction numerator 1 plus sin space straight theta over denominator cos space straight theta end fraction close square brackets space space space space space space space space space equals cot to the power of negative 1 end exponent open square brackets fraction numerator cos squared begin display style straight theta over 2 end style plus sin squared begin display style straight theta over 2 end style plus 2 sin begin display style straight theta over 2 end style cos begin display style straight theta over 2 end style over denominator cos squared straight theta over 2 minus sin squared straight theta over 2 end fraction close square brackets space space space space space space space space space equals cot to the power of negative 1 end exponent open square brackets fraction numerator open parentheses cos straight theta over 2 plus sin straight theta over 2 close parentheses squared over denominator open parentheses cos straight theta over 2 minus sin straight theta over 2 close parentheses open parentheses cos straight theta over 2 plus sin straight theta over 2 close parentheses end fraction close square brackets equals cot to the power of negative 1 end exponent open square brackets fraction numerator cos begin display style straight theta over 2 end style plus sin begin display style straight theta over 2 end style over denominator cos begin display style straight theta over 2 end style minus sin begin display style straight theta over 2 end style end fraction close square brackets space space space space space space space space space equals cot to the power of negative 1 end exponent open square brackets fraction numerator 1 plus tan straight theta over 2 over denominator 1 minus tan straight theta over 2 end fraction close square brackets equals cot to the power of negative 1 end exponent open square brackets cot open parentheses straight pi over 2 minus straight theta over 2 close parentheses close square brackets equals straight pi over 2 minus straight theta over 2 equals straight pi over 2 minus straight theta over 2 tan to the power of negative 1 end exponent straight x therefore dy over dx equals 0 plus 1 half cross times fraction numerator 1 over denominator 1 plus straight x squared end fraction equals fraction numerator 1 over denominator 2 left parenthesis 1 plus straight x squared right parenthesis end fraction

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