Question

$If space straight x equals 3 space sin space straight t minus sin space 3 straight t comma space straight y equals 3 space cos space straight t minus cos space 3 space straight t comma space find space dy over dx space at space straight t equals straight pi over 3.$

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Solution

space space space space space space space space space space space straight x equals 3 space sin space straight t minus sin space 3 straight t comma space space space space space straight y equals 3 space cos space straight t minus cos space 3 space straight t therefore space dx over dt equals 3 space cos space straight t minus 3 space cos space 3 straight t comma space dy over dt equals negative 3 space sin space straight t plus 3 space sin space 3 straight t Now space dy over dx equals fraction numerator begin display style dy over dt end style over denominator begin display style dx over dt end style end fraction equals fraction numerator negative 3 space sin space straight t plus 3 space sin space 3 straight t over denominator 3 space cos space straight t minus 3 space cios space 3 straight t end fraction equals fraction numerator sin space 3 straight t minus sin space straight t over denominator cos space straight t minus cos space 3 straight t end fraction equals fraction numerator 2 space cos begin display style fraction numerator 3 straight t plus straight t over denominator 2 end fraction end style sin begin display style fraction numerator 3 straight t minus straight t over denominator 2 end fraction end style over denominator 2 space sin space begin display style fraction numerator 3 straight t plus straight t over denominator 2 end fraction end style sin begin display style fraction numerator 3 straight t minus straight t over denominator 2 end fraction end style end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator cos space 2 straight t over denominator sin space 2 straight t end fraction equals cos space 2 straight t At space straight t equals straight pi over 3 comma space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative 2 over 3 fraction numerator cosec squared space begin display style fraction numerator 2 straight pi over denominator 3 end fraction end style over denominator cos begin display style straight pi over 3 end style minus cos space straight pi end fraction equals negative 2 over 3 fraction numerator cosec squared space begin display style straight pi over 3 end style over denominator cos begin display style straight pi over 3 end style minus cos space straight pi end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals negative 2 over 3. fraction numerator open parentheses begin display style fraction numerator 2 over denominator square root of 3 end fraction end style close parentheses squared over denominator begin display style 1 half end style plus 1 end fraction equals negative 2 over 3. fraction numerator begin display style 4 over 3 end style over denominator begin display style 3 over 2 end style end fraction equals negative 2 over 3 cross times 2 over 3 cross times 4 over 3 equals negative 16 over 27

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Continuity And Differentiability

$If space straight x equals 3 space sin space straight t minus sin space 3 straight t comma space straight y equals 3 space cos space straight t minus cos space 3 space straight t comma space find space dy over dx space at space straight t equals straight pi over 3.$

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