Question

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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Solution

Let space space straight y equals 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 Put space space straight x equals cos space straight theta therefore space space space space straight y equals sin to the power of negative 1 end exponent square root of fraction numerator 1 plus cos space straight theta over denominator 2 end fraction end root equals sin to the power of negative 1 end exponent square root of fraction numerator 2 cos squared begin display style straight theta over 2 end style over denominator 2 end fraction end root equals sin to the power of negative 1 end exponent open parentheses cos straight theta over 2 close parentheses space space space space space space space space space space equals sin to the power of negative 1 end exponent open square brackets sin 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 1 half cos to the power of negative 1 end exponent straight x therefore space dy over dx equals 0 minus 1 half fraction numerator negative 1 over denominator square root of 1 minus straight x squared end root end fraction equals fraction numerator 1 over denominator 2 square root of 1 minus straight x squared end root end fraction

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