A) \[\frac{x+y}{xy}\]
B) \[xy\]
C) \[\frac{x}{y}\]
D) \[\frac{y}{x}\]
Correct Answer: D
Solution :
Since, \[{{x}^{m}}.{{y}^{n}}=(x+y{{0}^{m+n}}\] Taking log on both sides, we get \[m\log x+n\log y=(m+n)\log (x+y)\] On differentiating w.r.t. x. we get \[\frac{m}{x}+\frac{n}{y}\frac{dy}{dx}=\frac{(m+n)}{(x+y)}\left( 1+\frac{dy}{dx} \right)\] \[\Rightarrow \]\[\frac{dy}{dx}\left( \frac{m+n}{x+y}-\frac{n}{y} \right)=\frac{m}{x}-\frac{m+n}{x+y}\] \[\Rightarrow \]\[\frac{dy}{dx}\left( \frac{my+ny-nx-ny}{y(x+y)} \right)=\frac{mx+my-mx-nx}{x(x+y)}\] \[\Rightarrow \]\[\frac{dy}{dx}=\frac{y}{x}\]
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