Question

$Find space dy over dx when space straight x equals straight a left parenthesis straight theta plus sinθ right parenthesis comma space straight y equals straight a left parenthesis 1 plus cosθ right parenthesis.$

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Solution

space space space space space space space space space space space space space space space straight x equals straight a left parenthesis straight theta plus sinθ right parenthesis space rightwards double arrow space dx over xθ equals straight a left parenthesis 1 plus cosθ right parenthesis space space space space space space space space space space space space space space space straight y equals straight a left parenthesis 1 plus cosθ right parenthesis space rightwards double arrow space dx over dθ equals negative straight a space sinθ Now space dy over dx equals fraction numerator begin display style dy over dθ end style over denominator begin display style dx over dθ end style end fraction equals fraction numerator negative straight a space sinθ over denominator straight a left parenthesis 1 plus cosθ right parenthesis end fraction equals fraction numerator negative 2 sin begin display style straight theta over 2 end style cos begin display style straight theta over 2 end style over denominator cos begin display style straight theta over 2 end style end fraction equals negative fraction numerator sin begin display style straight theta over 2 end style over denominator cos begin display style straight theta over 2 end style end fraction equals negative tan straight theta over 2

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