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Continuity And Differentiability

Question

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

Solution

Let space space space space space straight y equals sin to the power of negative 1 end exponent square root of fraction numerator 1 plus straight x squared over denominator 2 end fraction end root therefore space dy over dx equals fraction numerator 1 over denominator square root of 1 minus open parentheses square root of fraction numerator 1 plus straight x squared 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 squared 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 squared 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 squared end root right parenthesis space space space space space space space space space space space space space equals fraction numerator square root of 2 over denominator square root of 1 minus straight x squared end root end fraction. fraction numerator 1 over denominator square root of 2 end fraction. fraction numerator 2 straight x over denominator 2 square root of 1 minus straight x squared end root end fraction equals fraction numerator straight x over denominator square root of 1 minus straight x to the power of 4 end root end fraction

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