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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $square root of fraction numerator 1 minus tan space straight x over denominator 1 plus tan space straight x end fraction end root$

Solution

Let space straight y equals square root of fraction numerator 1 minus tan space straight x over denominator 1 plus tan space straight x end fraction end root equals square root of tan open parentheses straight pi over 4 minus straight x close parentheses end root equals open square brackets tan open parentheses straight pi over 4 minus straight x close parentheses close square brackets to the power of 1 half end exponent space dy over dx equals 1 half open square brackets tan open parentheses straight pi over 4 minus straight x close parentheses close square brackets to the power of negative 1 half end exponent. straight d over dx open square brackets tan open parentheses straight pi over 4 minus straight x close parentheses close square brackets space space space space space space space space space equals fraction numerator 1 over denominator 2 square root of tan open parentheses straight pi over 4 minus straight x close parentheses end root end fraction sec squared open parentheses straight pi over 4 minus straight x close parentheses. straight d over dx open parentheses straight pi over 4 minus straight x close parentheses space space space space space space space space space equals fraction numerator 1 over denominator 2 square root of tan open parentheses straight pi over 4 minus straight x close parentheses end root end fraction sec squared open parentheses straight pi over 4 minus straight x close parentheses cross times negative 1 equals negative fraction numerator sec squared open parentheses straight pi over 4 minus straight x close parentheses over denominator 2 square root of tan open parentheses straight pi over 4 minus straight x close parentheses end root end fraction

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