Question

Differentiate the following w.r.t.x: $sec to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator 2 straight x squared minus 1 end fraction close parentheses comma space 0 less than straight x less than fraction numerator 1 over denominator square root of 2 end fraction$

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Solution

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

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