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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $straight e to the power of cos minus 1 left parenthesis straight x plus 1 right parenthesis end exponent$

Solution

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

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