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Continuity And Differentiability

Question

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

Solution

Let space space space space straight y equals square root of fraction numerator sec minus 1 over denominator sec plus 1 end fraction end root equals square root of fraction numerator begin display style fraction numerator 1 over denominator cos space straight x end fraction minus 1 end style over denominator begin display style fraction numerator 1 over denominator cos space end fraction plus 1 end style end fraction end root equals square root of fraction numerator 1 minus cos space straight x over denominator 1 plus cos space straight x end fraction end root space space space space space space space space space space space space space equals square root of fraction numerator 2 sin squared begin display style straight x over 2 end style over denominator 2 cos squared begin display style straight x over 2 end style end fraction end root equals square root of tan squared straight x over 2 end root equals tan straight x over 2 therefore space dy over dx equals sec squared straight x over 2 cross times straight d over dx open parentheses straight x over 2 close parentheses equals sec squared straight x over 2 cross times 1 half equals 1 half sec squared straight x over 2

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