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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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Let space space space straight y equals 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 Put space space space straight x equals straight a space sin space straight theta space therefore space space space space space straight y equals tan to the power of negative 1 end exponent open parentheses fraction numerator straight a space sinθ over denominator square root of straight a squared minus straight a squared sin squared end root straight theta end fraction close parentheses tan to the power of negative 1 end exponent open parentheses fraction numerator straight a space sinθ over denominator straight a space cos space straight theta end fraction close parentheses equals tan to the power of negative 1 end exponent left parenthesis tan space straight theta right parenthesis equals straight theta therefore space space space space space straight y equals sin to the power of negative 1 end exponent open parentheses straight x over straight a close parentheses therefore space dy over dx equals fraction numerator 1 over denominator square root of 1 minus begin display style straight x squared over straight a squared end style end root end fraction

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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

88 Views