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Continuity And Differentiability

Question

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

Solution

Let space space space space space straight y equals cos to the power of negative 1 end exponent square root of fraction numerator 1 plus straight x over denominator 2 end fraction end root therefore space dy over dx equals negative fraction numerator 1 over denominator square root of 1 minus open parentheses square root of fraction numerator 1 plus straight x over denominator 2 end fraction end root close parentheses squared end root end fraction. straight d over dx open parentheses square root of fraction numerator 1 plus straight x over denominator 2 end fraction end root close parentheses equals fraction numerator 1 over denominator square root of 1 minus fraction numerator 1 plus straight x over denominator 2 end fraction end root end fraction. fraction numerator 1 over denominator square root of 2 end fraction straight d over dx left parenthesis square root of 1 plus straight x end root right parenthesis space space space space space space space space space space space space space equals negative fraction numerator square root of 2 over denominator square root of 1 minus straight x end root end fraction. fraction numerator 1 over denominator square root of 2 end fraction. fraction numerator 1 over denominator 2 square root of 1 plus straight x end root end fraction equals negative fraction numerator 1 over denominator 2 square root of 1 minus straight x squared end root end fraction

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