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 Multiple Choice QuestionsShort Answer Type

421. Differentiate space straight w. straight r. straight t. straight x colon space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator 1 plus 2 sin space straight x over denominator 2 plus cos space straight x end fraction close parentheses
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422. Differentiate space sin to the power of negative 1 end exponent open parentheses fraction numerator straight a plus straight b space cos space straight x over denominator straight b plus straight a space cos space straight x end fraction close parentheses space straight w. straight r. straight t. straight x comma space using space chain space rule
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423. Prove space that space straight d over straight d open square brackets square root of 1 minus straight x squared end root sin to the power of negative 1 end exponent straight x minus straight x close square brackets equals negative fraction numerator straight x space sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction


straight L. straight H. straight S. space space equals space straight d over dx open square brackets square root of 1 minus straight x squared end root sin to the power of negative 1 end exponent straight x minus straight x close square brackets equals straight d over dx open square brackets square root of 1 minus straight x squared end root sin to the power of negative 1 end exponent straight x close square brackets minus straight d over dx left parenthesis straight x right parenthesis
space space space space space space space space space space space space space equals square root of 1 minus straight x squared end root. straight d over dx left parenthesis sin to the power of negative 1 end exponent straight x right parenthesis plus sin to the power of negative 1 end exponent straight x. straight d over dx open parentheses square root of 1 minus straight x squared end root close parentheses minus 1
space space space space space space space space space space space space space equals square root of 1 minus straight x squared end root. fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction plus sin to the power of negative 1 end exponent straight x. fraction numerator negative 2 straight x over denominator 2 square root of 1 minus straight x squared end root end fraction minus 1 equals 1 minus fraction numerator straight x space sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction minus 1
space space space space space space space space space space space space space equals negative fraction numerator straight x space sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction equals straight R. straight H. straight S.
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424. Prove space that space straight d over dx left square bracket 2 straight x space tan to the power of negative 1 end exponent straight x minus log left parenthesis 1 plus straight x squared right parenthesis right square bracket equals 2 space tan to the power of negative 1 end exponent straight x.
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425. Prove space that space straight d over dx open square brackets straight x over 2 square root of straight a squared minus straight x squared end root plus straight a squared over 2 sin to the power of negative 1 end exponent straight x over straight a close square brackets equals square root of straight a squared minus straight x squared end root
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426. Differerntiate space cos to the power of negative 1 end exponent open parentheses fraction numerator 3 cos space straight x minus 4 sin space straight x over denominator 5 end fraction close parentheses space straight w. straight r. straight t. straight x.
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427. Find space dy over dx space if space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator 5 straight x plus 12 square root of 1 minus straight x squared end root over denominator 13 end fraction close parentheses
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428. If space straight y equals sec to the power of negative 1 end exponent open parentheses fraction numerator straight x plus 1 over denominator straight x minus 1 end fraction close parentheses plus sin to the power of negative 1 end exponent open parentheses fraction numerator straight x plus 1 over denominator straight x minus 1 end fraction close parentheses comma space find space dy over dx.
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429. Differentiate the following w.r.t.x: tan to the power of negative 1 end exponent open square brackets open parentheses fraction numerator 1 minus cos space straight x over denominator 1 plus cos space straight x end fraction close parentheses to the power of 1 half end exponent close square brackets
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430. Differentiate the following w.r.t.x: tan to the power of negative 1 end exponent open parentheses square root of fraction numerator 1 plus sin space straight x over denominator 1 minus sin space straight x end fraction end root close parentheses
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