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If $${x^m}.{y^n} = {\left( {x + y} \right)^{m + n}},$$ then $${{{dy} \over {dx}}}$$ is

JEE · Math · previous-year question

  1. A.$${y \over x}$$correct
  2. B.$${{x + y} \over {xy}}$$
  3. C.$$xy$$
  4. D.$${x \over y}$$

Answer

A. $${y \over x}$$

Explanation

$${x^m}.{y^n} = {\left( {x + y} \right)^{m + n}}$$ $$ \Rightarrow m\ln x + n\ln y = \left( {m + n} \right)\ln \left( {x + y} \right)$$ Differentiating both sides. $$\therefore$$ $${m \over x} + {n \over y}{{dy} \over {dx}} = {{m + n} \over {x + y}}\left( {1 + {{dy} \over {dx}}} \right)$$ $$ \Rightarrow \left( {{m \over x} - {{m + n} \over {x + y}}} \right) = \left( {{{m + n} \over {x + y}} - {n \over y}} \right){{dy} \over {dx}}$$ $$ \Rightarrow {{my - nx} \over {x\left( {x + y} \right)}} = \left( {{{my - nx} \over {y\left( {x + y} \right)}}} \right){{dy} \over {dx}}$$ $$ \Rightarrow {{dy} \over {dx}} = {y \over x}$$

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