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Continuity And Differentiability

Question
CBSEENMA12035141
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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

Solution
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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