Question

Differentiate the following w.r.t.x: $cos to the power of negative 1 end exponent open parentheses square root of fraction numerator 1 plus straight x over denominator 2 end fraction end root close parentheses$

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Solution

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

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