Question

$Prove space that space straight d over dx open square brackets straight x over 2 square root of straight a squared minus straight x squared end root plus straight a squared over 2 sin to the power of negative 1 end exponent straight x over straight a close square brackets equals square root of straight a squared minus straight x squared end root$

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Solution

straight L. straight H. straight S. space equals 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 space space space space space space space space space space space space space equals 1 half straight d over dx open parentheses straight x square root of straight a squared minus straight x squared end root close parentheses plus straight a squared over 2 straight d over dx open parentheses sin to the power of negative 1 end exponent straight x over straight a close parentheses space space space space space space space space space space space space space equals 1 half open square brackets straight x. fraction numerator negative 2 straight x over denominator 2 square root of straight a squared minus straight x squared end root end fraction plus square root of straight a squared minus straight x squared end root.1 close square brackets plus straight a squared over 2. fraction numerator 1 over denominator square root of 1 minus open parentheses begin display style straight x over straight a end style close parentheses squared end root end fraction.1 over straight a space space space space space space space space space space space space space equals 1 half open square brackets negative fraction numerator straight x squared over denominator square root of straight a squared minus straight x squared end root end fraction plus fraction numerator square root of straight a squared minus straight x squared end root over denominator 1 end fraction close square brackets plus fraction numerator straight a squared over denominator 2 square root of straight a squared minus straight x squared end root end fraction space space space space space space space space space space space space space equals 1 half open square brackets fraction numerator negative straight x squared plus straight a squared minus straight x squared over denominator square root of straight a squared minus straight x squared end root end fraction close square brackets plus fraction numerator straight a squared over denominator 2 square root of straight a squared minus straight x squared end root end fraction equals fraction numerator straight a squared minus 2 straight x squared over denominator 2 square root of straight a squared minus straight x squared end root end fraction plus fraction numerator straight a squared over denominator 2 square root of straight a squared minus straight x squared end root end fraction space space space space space space space space space space space space space equals fraction numerator straight a squared minus 2 straight x squared plus straight a squared over denominator 2 square root of straight a squared minus straight x squared end root end fraction equals fraction numerator 2 left parenthesis straight a squared minus straight x squared right parenthesis over denominator 2 square root of straight a squared minus straight x squared end root end fraction equals square root of straight a squared minus straight x squared end root equals straight R. straight H. straight S

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